Inclusion Criteria:

* Age ≥ 18 years
* Relapsed or refractory multiple myeloma who has had at least 3 prior lines of therapies including a proteasome inhibitor, immunomodulatory drug (IMiD) and anti-CD38 monoclonal antibody (mAb)
* Ciltacabtagene autoleucel (Carvykti) available for patient
* Measurable disease
* Eastern Cooperative Oncology Group (ECOG) performance status (PS) 0, 1 or 2
* Life expectancy ≥ 12 weeks
* Hemoglobin ≥ 8.0 g/dL (obtained ≤ 14 days prior to registration)
* Absolute neutrophil count (ANC) ≥ 1,000/mm\^3 (obtained ≤ 14 days prior to registration)
* Platelet count ≥ 50,000/mm\^3 (obtained ≤ 14 days prior to registration)
* Total bilirubin ≤ 1.5 x upper limit of normal (ULN) (obtained ≤ 14 days prior to registration)

* Note: Patients with Gilbert's syndrome must have a total bilirubin of ≤ 3 x ULN (obtained ≤ 14 days prior to registration)
* Alanine aminotransferase (ALT) and aspartate transaminase (AST) ≤ 2 x ULN (obtained ≤ 14 days prior to registration)
* Alkaline phosphatase ≤ 1.5 x ULN (obtained ≤ 14 days prior to registration)
* Calculated creatinine clearance ≥ 30 ml/min using the Cockcroft-Gault formula (obtained ≤ 14 days prior to registration)
* Negative pregnancy test done ≤ 7 days prior to registration, for persons of childbearing potential only

* Note: If the urine test is positive or cannot be confirmed as negative, a serum pregnancy test will be required
* Sexually active patients and their partners must use an effective method of contraception associated with a low failure rate prior to study entry and for the duration of study participation and for at least 30 days after the last dose of study drug
* Provide written informed consent
* Willingness to provide mandatory blood and bone marrow specimens for correlative research
* Willingness to provide mandatory bone marrow cores and/or tissue specimens for correlative research
* Willing to return to enrolling institution for follow-up (during the Active Monitoring Phase of the study)

Exclusion Criteria:

* Monoclonal gammopathy of undetermined significance, smoldering multiple myeloma, or AL amyloidosis
* Failure to recover from acute, reversible effects of prior therapy regardless of interval since last treatment.

* EXCEPTION: Grade 1 peripheral (sensory) neuropathy that has been stable for at least 1 month since completion of prior treatment
* Any of the following because this study involves an agent that has known genotoxic, mutagenic, and teratogenic effects:

* Pregnant persons
* Nursing persons
* Persons of childbearing potential (and persons able to father a child) who are unwilling to employ adequate contraception
* Major surgery ≤ 28 days prior to registration
* Co-morbid systemic illnesses or other severe concurrent disease which, in the judgment of the investigator, would make the patient inappropriate for entry into this study or interfere significantly with the proper assessment of safety and toxicity of the prescribed regimens
* Immunocompromised patients and patients known to be HIV positive.

* NOTE: Patients known to be HIV positive, but without clinical evidence of an immunocompromised state, or those currently receiving antiretroviral therapy with good control of HIV, are eligible for this trial
* Evidence of cardiovascular disease risk, as defined by any of the following:

* Evidence of current clinically significant uncontrolled arrhythmias, including clinically significant ECG abnormalities such as 2nd degree (Mobitz Type II) or 3rd degree atrioventricular (AV) block
* History of myocardial infarction, acute coronary syndromes (including unstable angina), coronary angioplasty, or stenting or bypass grafting within three (3) months of screening.
* Class III or IV heart failure as defined by the New York Heart Association functional classification system
* Uncontrolled hypertension
* History of life-threatening ventricular arrhythmias
* QTC interval \[electrocardiogram (ECG)\] ≥ 450 msec
* Uncontrolled intercurrent illness including, but not limited to:

* Ongoing or active infection
* Any medical condition that would make participation unduly hazardous
* Receiving any other investigational agent which would be considered as a treatment for the primary neoplasm
* Live vaccine ≤ 6 weeks prior to registration
* Has taken a strong inhibitor or inducer of CYP3A4/5, including grapefruit, St. John's Wort or related products ≤ 14 days prior to registration.

* Note: If required, patients may receive a short course of strong inhibitors or inducers for treatment of symptoms, but dasatinib dose must be adjusted as indicated
* Known hypersensitivity or allergy to dasatinib or quercetin
* Patients on therapeutic doses of anticoagulants (e.g. warfarin, heparin, low molecular weight heparin, factor Xa inhibitors, etc).
* On antiplatelet agents (e.g. full dose aspirin, clopidogrel etc.)

* NOTE: Baby aspirin, if necessary for cardioprotection, will be allowed
* On quinolone antibiotic therapy for treatment or for prevention of infections products ≤10 days prior to registration