This pilot phase II trial studies how well a new reduced intensity conditioning regimen that includes haploidentical donor NK cells followed by the infusion of selectively T-cell depleted progenitor cell grafts work in treating younger patients with hematologic malignancies that have returned after or did not respond to treatment with a prior transplant. Giving chemotherapy and natural killer cells before a donor progenitor cell transplant may help stop the growth of cells in the bone marrow, including normal blood-forming cells (progenitor cells) and cancer cells. It may also stop the patient's immune system from rejecting the donor's cells. When the healthy progenitor cells from a related donor are infused into the patient they make 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). Removing specific T cells from the donor cells before the transplant may prevent this.