Improving Leukemia or Lymphoma Treatment with Healthy People

This phase II trial studies how well cluster of differentiation 8 (CD8)+ memory T-cells work as a consolidative therapy following a donor non-myeloablative hematopoietic cell transplant in treating patients with leukemia or lymphoma. Giving total lymphoid irradiation and anti-thymocyte globulin before a donor hematopoietic cell transplant helps stop the growth of cells in the bone marrow, including normal blood-forming cells (stem cells) and cancer 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 make an immune response against the body's normal cells (called graft-versus-host disease). Giving cyclosporine and mycophenolate mofetil after the transplant may stop this from happening. Once the donated stem cells begin working, the patient's immune system may see the remaining cancer cells as not belonging in the patient's body and destroy them. Giving an infusion of the donor's white blood cells, such as CD8+ memory T-cells, may boost this effect and may be an effective treatment to kill any cancer cells that may be left in the body (consolidative therapy).