Inclusion Criteria:

* Documented informed consent of the participant and/or legally authorized representative

* Assent, when appropriate, will be obtained per institutional guidelines
* Agreement to allow the use of archival tissue from diagnostic tumor biopsies

* If unavailable, exceptions may be granted with study principal investigator (PI) approval
* Age: =\< 80 years

* Note: Patients \> 70 years of age must have Karnofsky performance status \>= 80 and HCT-comorbidity index (CI) =\< 2
* Karnofsky performance status \>= 70%
* Patients with the following diagnosis, eligible to undergo allogeneic HCT from an 8/8 match related/unrelated donor (A, B, C, DR by high resolution typing)

* Acute leukemias (acute myeloid leukemia \[AML\] or acute lymphoblastic leukemia \[ALL\]) in complete remission with bone marrow (BM) blast of \< 5%
* Myelofibrosis (MF): Primary or secondary with high- or intermediate-2 risk per Dynamic International Prognostic Scoring System (DIPSS)
* Myelodysplastic syndrome (blast \< 10%)
* Myeloproliferative neoplasm (MPN) other than MF needing HCT
* Chronic myelomonocytic leukemia (CMML)
* Total bilirubin =\< 2 x upper limit of normal (ULN) (unless has Gilbert's disease) (within 30 days prior to day 1 of protocol therapy unless otherwise stated)
* Serum glutamic-oxaloacetic transaminase (SGOT) and serum glutamate pyruvate transaminase (SGPT) \< 5 x ULN (within 30 days prior to day 1 of protocol therapy unless otherwise stated)
* Creatinine clearance of \>= 60 mL/min per 24 hour urine test or the Cockcroft-Gault formula (within 30 days prior to day 1 of protocol therapy unless otherwise stated)
* Left ventricular ejection fraction (LVEF) \>= 50%

* Note: To be performed within 30 days prior to day 1 of protocol therapy
* If able to perform pulmonary function tests: forced expiratory volume in 1 second (FEV1), forced vital capacity (FVC) and diffusion capacity of the lungs for carbon monoxide (DLCO) (diffusion capacity) \>= 50% of predicted (corrected for hemoglobin) (within 30 days prior to day 1 of protocol therapy)
* If unable to perform pulmonary function tests: O2 saturation \> 92% on room air (within 30 days prior to day 1 of protocol therapy)
* Seronegative for human immunodeficiency virus (HIV) antigen (Ag)/antibody (Ab) combo, hepatitis C virus (HCV), active hepatitis B virus (HBV) (surface antigen negative), and syphilis rapid plasma reagin (RPR) (within 30 days prior to day 1 of protocol therapy)

* If positive, hepatitis C ribonucleic acid (RNA) quantitation must be performed OR
* If seropositive for HIV, HCV or HBV, nucleic acid quantitation must be performed. Viral load must be undetectable
* Meets other institutional and federal requirements for infectious disease titer requirements

* Note: Infectious disease testing to be performed within 28 days prior to day 1 of protocol therapy
* Women of childbearing potential (WOCBP): negative urine or serum pregnancy test (within 30 days prior to day 1 of protocol therapy)

* If the urine test is positive or cannot be confirmed as negative, a serum pregnancy test will be required
* Agreement by females and males of childbearing potential to use an effective method of birth control or abstain from heterosexual activity for the course of the study through at least 6 months after the last dose of protocol therapy

* Childbearing potential defined as not being surgically sterilized (men and women) or have not been free from menses for \> 1 year (women only)

Exclusion Criteria:

* Prior allogeneic HCT
* Chemotherapy, radiation therapy, biological therapy, immunotherapy within 21 days prior to day 1 of protocol therapy

* Note: Conditioning regimen within 21 days prior to day 1 of protocol therapy is not considered as an exclusion criterion. Patients on maintenance chemotherapy are not excluded
* Other investigational drugs for treatment of GVHD
* History of allergic reactions attributed to compounds of similar chemical or biologic composition to study agents
* Psychological issues, no appropriate caregivers identified, or non-compliant to medication
* Clinically significant uncontrolled illness
* Uncontrolled infection (bacterial, viral, fungal)
* Other active malignancy
* Females only: Pregnant or breastfeeding
* Any other condition that would, in the investigator's judgment, contraindicate the patient's participation in the clinical study due to safety concerns with clinical study procedures
* Prospective participants who, in the opinion of the investigator, may not be able to comply with all study procedures (including compliance issues related to feasibility/logistics)