This phase II trial is studying how well fludarabine phosphate and total-body irradiation followed by donor peripheral blood stem cell transplant work in treating patients with acute lymphoblastic leukemia or chronic myelogenous leukemia that has responded to previous treatment with imatinib mesylate, dasatinib, or nilotinib. Giving low doses of chemotherapy, such as fludarabine phosphate, and total-body irradiation (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. The donated stem cells may replace the patient's immune system and help destroy any remaining cancer cells (graft-versus-tumor effect). Giving an infusion of the donor's T cells (donor lymphocyte infusion) after the transplant may help increase this effect. Sometimes the transplanted cells from a donor can also make an immune response against the body's normal cells. Giving mycophenolate mofetil and cyclosporine after the transplant may stop this from happening.