Donor Peripheral Blood Stem Cell Transplant in Treating Patients With Hematologic Malignancies

PRIMARY OBJECTIVES: I. To demonstrate that use of PBSC in place of marrow as the source of lymphocytes and stem cells for nonmyeloablative transplants from related, haploidentical donors will not result in unacceptable rates of high-grade acute or chronic GVHD, non-relapse mortality or relapse compared to historical data on nonmyeloablative transplants from unrelated donors. SECONDARY OBJECTIVES: I. Estimates of the rates of neutrophil and platelet recovery, number of red blood cell (RBC) and platelet transfusions, incidences of graft failure, transplant-related toxicities, disease-free survival and overall survival. OUTLINE: Patients receive fludarabine intravenously (IV) over 30-60 minutes daily on days -6 through -2 and cyclophosphamide IV over 1-2 hours on days -6, -5, and 3-4. Patients undergo total-body irradiation on day -1. Patients undergo donor peripheral blood stem cell transplant on day 0. Patients then receive tacrolimus IV once daily or orally (PO) twice daily (BID) on days 5-180 (may be continued if active GVHD is present), mycophenolate mofetil IV or PO thrice daily (TID) on days 5-35 (may be continued if GVHD present), and filgrastim IV beginning on day 5 until the absolute neutrophil count (ANC) is \>= 1,000/mm\^3 for three consecutive days. Treatment continues in the absence of disease progression or unacceptable toxicity.