This clinical trial studies fludarabine phosphate, low-dose total body irradiation, and donor stem cell transplant in treating patients with hematologic malignancies or kidney cancer. Giving chemotherapy drugs, such as fludarabine phosphate, and total-body irradiation 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 cells 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 cyclosporine before the transplant and cyclosporine and mycophenolate mofetil after the transplant may stop this from happening.