Provision of TCRγδ T Cells and Memory T Cells Plus Selected Use of Blinatumomab in Naïve T-cell Depleted Haploidentical Donor Hematopoietic Cell Transplantation for Hematologic Malignancies Relapsed or Refractory Despite Prior Transplantation

This study seeks to examine treatment therapy that will reduced regimen-related toxicity and relapse while promoting rapid immune reconstitution with limited serious graft-versus-host-disease (GVHD) and also improve disease-free survival and quality of life. The investigators propose to evaluate the safety and efficacy of selective naive T-cell depleted (by TCRɑβ and CD45RA depletion, respectively) haploidentical hematopoietic cell transplant (HCT) following reduced intensity conditioning regimen that avoids radiation in patients with hematologic malignancies that have relapsed or are refractory following prior allogeneic transplantation. PRIMARY OBJECTIVE: * To estimate engraftment by day +30 post-transplant in patients who receive TCRɑβ-depleted and CD45RA-depleted haploidentical donor progenitor cell transplantation following reduced intensity conditioning regimen without radiation. SECONDARY OBJECTIVES: * Assess the safety and feasibility of the addition of Blinatumomab in the early post-engraftment period in patients with CD19+ malignancy. * Estimate the incidence of malignant relapse, event-free survival, and overall survival at one-year post-transplantation. * Estimate incidence and severity of acute and chronic (GVHD). * Estimate the rate of transplant related mortality (TRM) in the first 100 days after transplantation.

Blood progenitor cells will be obtained from a partially matched adult family member (donor). After processing and filtration using the CliniMACS device, cells will be infused into participants meeting eligibility criteria. Prior to transplant, participants will receive a conditioning treatment of rabbit ATG, cyclophosphamide, fludarabine, thiotepa, melphalan, and rituximab. Mesna will be given to help prevent side effects of cyclophosphamide. Tacrolimus will be given to help reduce the risk of GVHD. G-CSF will be given after transplant to help the donor progenitor cells make white blood cells faster so that the immune system is better able to fight infection. Blood progenitor cells will be given in two infusions on Day 0 and Day +1. Progenitor cells then move through the blood stream to the bone marrow space where they should begin to grow. Participant blood will be monitored for 100 days to assure that the progenitor cells begin to grow. If the growth is low, additional progenitor cells may be given. Blood tests will be monitored for up to one year to observe how well the donor cells grow and their effect on the infection-fighting system.