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 pyruva…