This phase II trial is studying how well giving fludarabine and cyclophosphamide together with total-body irradiation and rituximab works in treating patients with B-cell lymphoma or chronic lymphocytic leukemia who are undergoing an allogeneic (donor) bone marrow transplant. The type of bone marrow transplant is a less intensive or "mini" transplant using a relative as the bone marrow donor. The donated bone marrow stem cells may replace the patient's immune system cells and help destroy any remaining cancer (graft-versus-tumor effect). Patients undergoing this type of transplant often have more than one relative who could be a donor. The trial is also studying a new way of choosing amongst possible donors which might improve how the rituximab works.