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

This phase II trial studies the side effect of busulfan, fludarabine phosphate, and post-transplant cyclophosphamide in treating patients with blood cancer undergoing donor stem cell transplant. Drugs used in chemotherapy, such as busulfan, fludarabine phosphate and cyclophosphamide work in different ways to stop the growth of cancer cells, either by killing the cells, by stopping them from dividing, or by stopping them from spreading. Giving chemotherapy such as busulfan and fludarabine phosphate before a donor stem cell transplant helps stop the growth of cells in the bone marrow, including normal blood-forming cells (stem cells) and cancer cells. Sometimes the transplanted cells from a donor can make an immune response against the body's normal cells (called graft-versus-host disease). Giving cyclophosphamide after the transplant may stop this from happening. Once the donated stem cells begin working, the patient's immune system may see the remaining cancer cells as not belonging in the patient's body and destroy them.

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