This phase II trial studies the side effects and the best dose of alemtuzumab when given together with fludarabine phosphate and low-dose total body irradiation (TBI) and how well it works before donor stem cell transplant in treating patients with hematological malignancies. Giving chemotherapy and low-dose TBI before a donor peripheral blood stem cell transplant helps stop the growth of cancer cells. It may also stop the patient's immune system from rejecting the donor's stem cells. Also, monoclonal antibodies, such as alemtuzumab, can find cancer cells and either kill them or deliver cancer-killing substances to them without harming normal cells. When the healthy stem cells from a 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 cells from a donor can also make an immune response against the body's normal cells. Giving cyclosporine (CSP) and mycophenolate mofetil (MMF) after transplant may stop this from happening.