This phase II for relapsed or refractory B-cell malignancies builds on the platform of nonmyeloablative, related-donor, HLA-matched or HLA-haploidentical BMT with post-transplantation high-dose cyclosphosphamide administered for prophylaxis of graft-versus-host disease and graft rejection. Rituximab is added to the transplant regimen with the goal of augmenting anti-tumor activity. In patients with B-cell lymphomas, specific polymorphisms in the immunoglobulin Fc receptor have been associated with greater sensitivity to rituximab or rituximab-based therapies, translating in some series into higher response rates and improved progression-free survival. This raises the possibility of selecting donors who carry this permissive polymorphism. This trial identifies and selects donors who have the favorable polymorphism at FcgammaR3A-158, thereby potentially conferring greater sensitivity to rituximab in the host after BMT.