This phase II trial studies how well giving fludarabine phosphate, melphalan, and low-dose total-body irradiation (TBI) followed by donor peripheral blood stem cell transplant (PBSCT) works in treating patients with hematologic malignancies. Giving chemotherapy drugs such as fludarabine phosphate and melphalan, and low-dose TBI before a donor PBSCT helps stop the growth of cancer and abnormal cells and helps stop the patient's immune system from rejecting the donor's stem cells. When the healthy stem cells from the donor are infused into the patient they may help the patient's bone marrow make stem cells, red blood cells, white blood cells, and platelets. Sometimes the transplanted cell from a donor can make an immune response against the body's normal cells. Giving tacrolimus, mycophenolate mofetil (MMF), and methotrexate after transplant may stop this from happening