Busulfan, Fludarabine Phosphate, and Post-Transplant Cyclophosphamide in Treating Patients With Blood Cancer Undergoing Donor Stem Cell Transplant

Inclusion Criteria:

* Patients with high-risk hematologic malignancies with anticipated poor prognosis with non transplant therapy, including those in remission or with induction failure and after treated or untreated relapse; Diagnoses to be included a) Acute myeloid leukemia; b) Acute lymphocytic leukemia; c) Chronic myeloid leukemia; d) Chronic lymphoproliferative disorder; e) Myelodysplastic syndrome; f) Myeloproliferative syndromes; g) Non-Hodgkin's lymphoma; h) Hodgkin's Lymphoma; i) Multiple myeloma
* Patients must have a haploidentical related donor or a fully matched related or unrelated donor
* Performance score of \>= 70 by Karnofsky/Lansky or performance score (PS) 0 to 1 (Eastern Cooperative Oncology Group \[ECOG\] =\< 1)
* Left ventricular ejection fraction \>= 50%
* Adequate pulmonary function with forced expiratory volume in 1 second (FEV1), forced vital capacity (FVC) and diffusion capacity of the lung for carbon monoxide (DLCO) \>= 50% of expected corrected for hemoglobin and/or volume; children unable to perform pulmonary function tests (e.g., less than 7 years old) pulse oximetry of \>= 92% on room air
* Creatinine clearance (calculated creatinine clearance by Cockcroft-Gault using adjusted body weight if actual body weight is 20% greater than ideal is permitted) should be \> 50 ml/min
* Bilirubin =\< 2 x the upper limit of normal (except with patients high indirect bilirubin due to Gilbert's syndrome, hypersplenism, or hemolysis)
* Serum glutamate pyruvate transaminase (SGPT) (alanine aminotransferase \[ALT\]) \< 200
* Negative beta human chorionic gonadotropin (HCG) test in a woman with child bearing potential, defined as not post-menopausal for 12 months or no previous surgical sterilization; women of child bearing potential must be willing to use an effective contraceptive measure while on study
* Patient or patient's legal representative able to sign informed consent

Exclusion Criteria:

* Human immunodeficiency virus (HIV) seropositivity
* Uncontrolled infections
* Patients with comorbidity score \> 3; the principal investigator is the final arbiter of eligibility for comorbidity score \> 3
* Prior allogeneic transplant
* Patients with active hepatitis B and C
* Patients with prior coronary artery disease
* Patients who received inotuzumab and/or gemtuzumab in the past