Inclusion Criteria:

* STEP 1: Participants must have systemic AL amyloidosis which is biopsy proven and includes histologically-confirmed by positive Congo red stain with green birefringence on polarized light microscopy, OR characteristic appearance by electron microscopy AND confirmatory AL amyloid typing (mass spectrometry-based proteomic analysis or immunofluorescence). If there is question regarding diagnosis, consult study chairs prior to registration
* STEP 1: Participants must have measurable disease within 28 days prior to treatment if initiated prior to registration or within 28 days of registration as defined by at least one of the following:

* Positive monoclonal serum immunofixation electrophoresis
* Positive monoclonal urine immunofixation electrophoresis
* Monoclonal plasma cells in bone marrow In addition, participants must also have a difference between the involved and uninvolved free light chain (dFLC) \>= 2 mg/dL
* STEP 1: Participants may receive up to one cycle (or 28 days) of therapy prior to enrollment. If a patient receives \>= 75% of 1 cycle of protocol identical Dara-VCD, this will be considered 1 cycle of protocol induction. Any patient who receives less than 75% of 1 cycle of Dara-VCD or non-protocol therapy will still be eligible but will be treated per protocol. If protocol identical therapy is initiated prior to enrollment, this treatment is not continued but rather treatment is dictated per protocol
* STEP 1: Participants may be receiving chronic corticosteroids if they are being given for disorders other than AL amyloidosis or myeloma
* STEP 1: Participant must be \>= 18 years old
* STEP 1: Participant must have Eastern Cooperative Oncology Group (ECOG) performance score (PS) of 0, 1, or 2 (PS = 3 may be allowed if secondary to neuropathy)
* STEP 1: Participant must have a complete medical history and physical exam within 28 DAYS prior to registration
* STEP 1: Participants must be willing to undergo high dose chemotherapy and autologous stem cell transplantation if they are randomized to the arm receiving high dose chemotherapy and autologous stem cell transplantation
* STEP 1: Participants must be eligible to receive high dose chemotherapy with melphalan at a dose of 200 mg/m\^2 or 140 mg/m\^2 (200 mg/m\^2 is highly encouraged but not mandated). Transplant eligibility criteria are included in the general eligibility criteria listed below:

* Participant must have a supine systolic blood pressure (BP) \>= 90 mmHg (at registration step-1, this may by supported by midodrine
* Participant must have non-severe cardiac AL (meeting all the below criteria) as defined by:

* N-terminal proB-type natriuretic peptide (NT proBNP) \< 5000 (if no NTproBNP, brain natriuretic peptide \[BNP\] must be available and \< 400)
* Troponin T (TnT) \< 0.06. If not available, one of the following two criteria must be met:

* High sensitivity troponin (hsTnT) T \< 75 or troponin I \< 0.1ng/dL
* New York Heart Association (NYHA) I or II
* Cardiac ejection fraction (EF) \>= 40%
* STEP 1: Hemoglobin \>= 8.0 g/dL (\> 5 mmol/L); red blood cell transfusion allowed up to 7 day prior to registration (within 28 days prior to registration) (NOTE: Growth factor support granulocyte colony-stimulating factor \[G-CSF\] is permitted per institutional guidelines)
* STEP 1: Leukocytes \>= 2 x 10\^3/uL (within 28 days prior to registration) (NOTE: Growth factor support \[G-CSF\] is permitted per institutional guidelines)
* STEP 1: Absolute neutrophil count \>= 1.0 x 10\^3/uL (within 28 days prior to registration) (NOTE: Growth factor support \[G-CSF\] is permitted per institutional guidelines)
* STEP 1: Platelets \>= 50 x 10\^3/uL (within 28 days prior to registration) (NOTE: Growth factor support \[G-CSF\] is permitted per institutional guidelines)
* STEP 1: Total bilirubin =\< 1.5 times the institutional upper limit of normal (ULN) unless history of Gilbert's disease. Participants with history of Gilbert's disease must have total bilirubin =\< 5 x institutional ULN (within 28 days prior to registration)
* STEP 1: Direct bilirubin =\< 2.0 mg/dL (within 28 days prior to registration)
* STEP 1: Aspartate aminotransferase (AST)/ alanine aminotransferase (ALT) =\< 3x upper limit of normal (ULN) (except if secondary to hepatic involvement) (within 28 days prior to registration)
* STEP 1: Alkaline phosphatase =\< 750 U/L (except if secondary to hepatic involvement) (within 28 days prior to registration)
* STEP 1: Participants must have a serum creatinine =\< the institutional (I)ULN OR measured OR calculated creatinine clearance \>= 30 mL/min using the following Cockcroft-Gault formula. This specimen must have been drawn and processed within 28 days prior to registration
* STEP 1: If peripheral neuropathy is present at diagnosis, participants must be grade 2 (moderate symptoms; limiting instrumental activity of daily living \[ADL\]) or less
* STEP 1: Participants must have adequate cardiac function. Participants with known history or current symptoms of cardiac disease, or history of treatment with cardiotoxic agents, must have a clinical risk assessment of cardiac function using the New York Heart Association Functional Classification. To be eligible for this trial, participants must be class 2 or better
* STEP 1: Participants must not be seropositive for hepatitis B (defined by a positive test for hepatitis B surface antigen \[HBsAg\]). Subjects with resolved infection (i.e., subjects who are HBsAg negative but positive for antibodies to hepatitis B core antigen \[anti-HBc\] and/or antibodies to hepatitis B surface antigen \[anti-HBs\]) must be screened using real-time polymerase chain reaction (PCR) measurement of hepatitis B virus (HBV) deoxyribonucleic acid (DNA) levels. Those who are PCR positive will be excluded. EXCEPTION: Subjects with serologic findings suggestive of HBV vaccination (anti-HBs positivity as the only serologic marker) AND a known history of prior HBV vaccination, do not need to be tested for HBV DNA by PCR
* STEP 1: Participants with a history of hepatitis C virus (HCV) infection must have been treated and cured. Participants currently being treated for HCV infection must have undetectable HCV viral load test on the most recent test results obtained within 6 months prior to registration, if indicated
* STEP 1: Participants must not have concurrent multiple myeloma as defined by the presence of lytic bone disease, plasmacytomas, \>= 60% plasma cells in the bone marrow, or hypercalcemia. Participants will not be excluded solely based on the presence of plasma cells \> 10% in the bone marrow unless the plasma cell percentage exceeds \>60%
* STEP 1: Participants must not have known allergies to any of the study drugs
* STEP 1: Participants must not have had a major surgery within 14 days prior to registration and be fully recovered from surgery completed within 14 days prior to registration
* STEP 1: Participants must not have a known chronic obstructive pulmonary disease with a forced expiratory volume in 1 second (FEV1) \< 50% of predicted normal
* STEP 1: Participants with known human immunodeficiency virus (HIV)-infection must be on effective anti-retroviral therapy at registration and have undetectable viral load test on the most recent test results obtained within 6 months prior to registration
* STEP 1: Participants must not have either moderate or severe persistent asthma within the past 2 years), or currently have uncontrolled asthma of any classification. (Note that subjects who currently have controlled intermittent asthma or controlled mild persistent asthma are allowed in the study)
* STEP 1: Participants must not have uncontrolled diabetes within 28 days prior to registration
* STEP 1: Participants must not have uncontrolled blood pressure and hypertension within 14 days prior to registration. Participants must have a supine systolic BP of \>= 90 mmHg
* STEP 1: Participants must not have a prior or concurrent malignancy whose natural history or treatment (in the opinion of the treating physician) has the potential to interfere with the safety or efficacy assessment of the investigational regimen
* STEP 1: Participants must not have received vaccination with live attenuated vaccines within 28 days prior to Registration to Step 1
* STEP 1: Participants must not have uncontrolled infection at the discretion of the enrolling physician and to be discussed with the study chair if the participant is on active anti-infectious therapy. Any patient on active anti-microbial therapy for chronic infectious issues should be discussed with the study chair prior to enrollment
* STEP 1: Participants must not be pregnant or nursing (nursing includes breast milk fed to an infant by any means, including from the breast, milk expressed by hand, or pumped). Individuals who are of reproductive potential must have agreed to use an effective contraceptive method with details provided as a part of the consent process. A person who has had menses at any time in the preceding 12 consecutive months or who has semen likely to contain sperm is considered to be of "reproductive potential." In addition to routine contraceptive methods, "effective contraception" also includes refraining from sexual activity that might result in pregnancy and surgery intended to prevent pregnancy (or with a side-effect of pregnancy prevention) including hysterectomy, bilateral oophorectomy, bilateral tubal ligation/occlusion, and vasectomy with testing showing no sperm in the semen
* STEP 1: Participants must be offered the opportunity to participate in specimen banking. With participant consent, specimens must be collected and submitted via the Southwestern Oncology group (SWOG) Specimen Tracking System
* STEP 1: Participants must agree to have blood, bone marrow core biopsy and aspirate, and fat pad biopsy specimens submitted for minimal residual disease assessment and future exploratory studies
* STEP 1: Participants who can complete PRO, QOL, PRO-CTCAE questionnaires, etc. forms in English, Spanish and French must participate in the patient-reported outcomes and quality of life
* STEP 2: Participants must have met all eligibility criteria for Step-1 registration
* STEP 2: Participants must have achieved at least a partial response
* STEP 2: Participants must continue receiving at least one of study drugs (bortezomib, cyclophosphamide, or daratumumab and hyaluronidase-fihj) if another study drug (daratumumab and hyaluronidase-fihj, cyclophosphamide, or bortezomib) has been discontinued due to adverse events. Note: daratumumab and hyaluronidase-fihj cannot be permanently discontinued
* STEP 2: Participants must have completed induction therapy
* STEP 2: Participants must be registered to Step 2 within 42 days of cycle 3, day 28 of induction therapy
* STEP 2: Participants must plan to initiate their assigned consolidation therapy within 8 weeks after randomization
* STEP 2: Participants must not have experienced a MOD-PFS event
* STEP 2: Participants must have ECOG performance score (PS) of 0, 1, or 2 (PS = 3 may be allowed if secondary to neuropathy)
* STEP 2: Participant must have a complete medical history and physical exam within 28 days prior to registration
* STEP 2: Participants must be willing to undergo high dose chemotherapy and autologous stem cell transplantation if they are randomized to the arm receiving high dose chemotherapy and autologous stem cell transplantation
* STEP 2: Participants randomized to Arm 2 must be willing and able to return to a participating treatment center for their assigned treatment after transplant. Note that participants need not to have a direct relationship with the transplant center in order to register
* STEP 2: Participants must be eligible to receive high dose chemotherapy with melphalan at a dose of 200 mg/m\^2 or 140 mg/m\^2 (200 mg/m\^2 is highly encouraged but not mandated). Transplant eligibility criteria are included in the general eligibility criteria listed below:

* Patient must have a supine systolic BP \>= 90 mmHg (at registration step-1, this may not by supported by midodrine)
* Patient must have non-severe cardiac AL as defined by:

* NT proBNP \<5000 (if no NTproBNP, BNP must be available and \< 400 pg/mL) (within 14 days prior to registration step-2)
* TnT \< 0.06. If not available, one of the following two criteria must be met (within 14 days prior to registration step-2)

* hsTnT \<75 or troponin I \< 0.1ng/dL
* NYHA I or II (within 14 days prior to registration step-2)
* Cardiac EF \>= 40% (within 14 days prior to registration step-2)
* STEP 2: Hemoglobin \> 8.0 g/dL (\> 5 mmol/L); red blood cell transfusion allowed up to 7 days prior to registration (within 28 days prior to registration) (NOTE: Growth factor support \[G-CSF\] is permitted per institutional guidelines)
* STEP 2: Leukocytes \>= 2 x 10\^3/uL (within 28 days prior to registration) (NOTE: Growth factor support \[G-CSF\] is permitted per institutional guidelines)
* STEP 2: Absolute neutrophil count \>= 1.0 x 10\^3/uL (within 28 days prior to registration) (NOTE: Growth factor support \[G-CSF\] is permitted per institutional guidelines)
* STEP 2: Platelets \>= 50 x 10\^3/uL (within 28 days prior to registration) (NOTE: Growth factor support \[G-CSF\] is permitted per institutional guidelines)
* STEP 2: Total bilirubin =\< 1.5 times the institutional ULN unless history of Gilbert's disease. Participants with history of Gilbert's disease must have total bilirubin =\< 5 x institutional ULN (within 28 days prior to registration)
* STEP 2: Direct bilirubin =\< 2.0 mg/dL (except if secondary to hepatic involvement) (within 28 days prior to registration)
* STEP 2: AST/ALT =\< 3x upper limit of normal (ULN) (except if secondary to hepatic involvement) (within 28 days prior to registration)
* STEP 2: Alkaline phosphatase =\< 750 U/L (except if secondary to hepatic involvement) (within 28 days prior to registration)
* STEP 2: Participants must have a serum creatinine =\< the IULN OR calculated creatinine clearance ≥ 30 mL/min using the following Cockcroft-Gault Formula. This specimen must have been drawn and processed within 28 days prior to registration
* STEP 2: Participants must not have uncontrolled infection at the discretion of the enrolling physician and to be discussed with the study chair if the participant is on active anti-infectious therapy. Any patient on active anti-microbial therapy for chronic infectious issues should be discussed with the study chair prior to enrollment
* STEP 2: Participants randomized to the ASCT arm must be able to have at least 2.0 x 10\^6 CD34 cells/kg collected
* STEP 2: Participants who can complete PRO, QOL, PRO-CTCAE questionnaires, etc. forms in English, Spanish and French must participate in the patient-reported outcomes and quality of life
* STEP 3: Participants must have met all eligibility criteria for Step-1 and Step-2 registration
* STEP 3: Participants must not have had daratumumab and hyaluronidase-fihj permanently discontinued during induction or consolidation
* STEP 3: Participants must have completed induction and consolidation therapy
* STEP 3: Participants must be registered to Step 3 within the following time frames:

* If randomized to Arm 1 Dara-VCD consolidation: within 28 days of completion of 3 cycles of consolidation therapy
* If randomized to Arm 2 high dose chemotherapy and autologous stem cell transplantation: within 180 days following initiation of stem cell transplantation
* STEP 3: Participants must not have experienced a MOD-PFS event
* STEP 3: Participants must have ECOG performance score (PS) of 0, 1, or 2 (PS = 3 is allowed if secondary to neuropathy)
* STEP 3: Participants must have a complete medical history and physical exam within 28 DAYS prior to registration
* STEP 3: Hemoglobin \> 8.0 g/dL (\> 5 mmol/L); red blood cell transfusion allowed up to 7 days prior to registration (within 28 days prior to registration) (NOTE: Growth factor support \[G-CSF\] is permitted per institutional guidelines)
* STEP 3: Leukocytes \>= 2 x 10\^3/uL (within 28 days prior to registration) (NOTE: Growth factor support \[G-CSF\] is permitted per institutional guidelines)
* STEP 3: Absolute neutrophil count \>= 1.0 x 10\^3/uL (within 28 days prior to registration) (NOTE: Growth factor support \[G-CSF\] is permitted per institutional guidelines)
* STEP 3: Platelets \>= 50 x 10\^3/uL (within 28 days prior to registration) (NOTE: Growth factor support \[G-CSF\] is permitted per institutional guidelines)
* STEP 3: Total bilirubin =\< 1.5 times the institutional upper limit of normal (ULN) unless history of Gilbert's disease. Participants with history of Gilbert's disease must have total bilirubin =\< 5 x institutional ULN (within 28 days prior to registration)
* STEP 3: Direct bilirubin =\< 2.0 mg/dL (within 28 days prior to registration)
* STEP 3: AST/ALT =\< 3x upper limit of normal (ULN) (except if secondary to hepatic involvement) (within 28 days prior to registration)
* STEP 3: Alkaline phosphatase =\< 750 U/L (except if secondary to hepatic involvement) (within 28 days prior to registration)
* STEP 3: Participants must not have uncontrolled infection at the discretion of the enrolling physician and to be discussed with the study chair if the participant is on active anti-infectious therapy. Any patient on active anti-microbial therapy for chronic infectious issues should be discussed with the study chair prior to enrollment
* Participants must be informed of the investigational nature of this study and must sign and give informed consent in accordance with institutional and federal guidelines.

For participants with impaired decision-making capabilities, legally authorized representatives may sign and give informed consent on behalf of study participants in accordance with applicable federal, local, and Central Institutional Review Board (CIRB) regulations