Comparing ATG or Post-Transplant Cyclophosphamide to Calcineurin Inhibitor-Methotrexate as GVHD Prophylaxis After Myeloablative Unrelated Donor Peripheral Blood Stem Cell Transplantation

OUTLINE: CONDITIONING REGIMENS: Participants receive 1 of 3 regimens and are randomized to 1 of 3 arms for GVHD prophylaxis. REGIMEN A: Participants undergo total body irradiation (TBI) twice daily (BID) on days -6 to -4 (-7 to -4 for those \< 18 years), then receive cyclophosphamide IV over 1-2 hours on days -3 and -2. Participants randomized to Arm 2 only receive TBI on days -3 to -1 (-4 to -1 for those \< 18 years). REGIMEN B: Participants receive fludarabine phosphate IV and busulfan IV every 6 hours on days -5 to -2. REGIMEN C: Participants receive busulfan orally (PO) or IV every 6 hours on days -7 to -4 and cyclophosphamide IV over 1-2 hours on days -3 and -2. Myelofibrosis or other myeloproliferative neoplasms: Participants \>= 18 years receive cyclophosphamide IV over 1-2 hours on days -7 and -6 and busulfan IV on days -5 to -2. Participants \< 17 years receive busulfan IV every 6 hours on days -7 to -4 and cyclophosphamide IV on days -3 and -2. All participants undergo peripheral blood stem cell transplantation on day 0. ARM 1: Participants receive anti-thymocyte globulin IV over 4-6 hours on days -3 to -1. Beginning day -1, participants also receive tacrolimus IV or cyclosporine IV twice daily (BID) tapered at day 50, and methotrexate IV on days 1, 3, 6 and 11 in the absence of disease progression or unacceptable toxicity. ARM 2: Participants receive cyclophosphamide IV over 1-2 hours on days 3 and 4. Beginning day 5, participants also receive tacrolimus IV or cyclosporine IV BID tapered at day 50 in the absence of disease progression or unacceptable toxicity. ARM 3: Beginning day -1, participants receive tacrolimus IV or cyclosporine IV BID tapered at day 50, and methotrexate IV on days 1, 3, 6 and 11 in the absence of disease progression or unacceptable toxicity. After completion of study treatment, participants are followed up at 6 months, then annually up to 5 years.