Brief Summary: This phase I/II trial studies the side effects and best dose of bendamustine when given with or without cyclophosphamide in preventing graft versus host disease (GVHD) in patients undergoing stem cell transplant. Drugs used in chemotherapy, such as bendamustine and cyclophosphamide, work in different ways to stop the growth of cancer cells, either by killing the cells, by stopping them from dividing, or by stopping them from spreading. Giving chemotherapy and total body irradiation before or after a stem cell transplant helps kills cancer cells that are in the body and helps make room in the patient's bone marrow for new blood-forming cells (stem cells) to grow. Sometimes, the transplanted cells from a donor can attack the body's normal cells called GVHD. Giving tacrolimus, mycophenolate mofetil, and filgrastim after the transplant may stop this from happening.

Detailed Description: PRIMARY OBJECTIVE: I. Evaluate the safety of substituting the standard post-transplant cyclophosphamide (PT-CY) given on day +3 and +4 with post-transplant bendamustine (PT-BEN) in patients undergoing HLA-mismatched hematopoietic cell transplantation. SECONDARY OBJECTIVES: I. To evaluate treatment-related mortality. II. To assess acute and chronic graft-versus-host disease (GVHD). III. To assess overall survival, progression-free survival and relapse rates. IV. To evaluate the risk of acute cystitis. V. To evaluate immune reconstitution after transplantation. OUTLINE: This is a dose-escalation study of bendamustine. Patients are assigned to 1 of 2 treatment schedules. SCHEDULE I (NON-LYMPHOMA): Patients receive fludarabine intravenously (IV) over 1 hour on days -5 to -2, melphalan IV over 30 minutes on days -5 and -4, and undergo total body irradiation (TBI) on day -1 and stem cell transplantation IV over 2-6 hours on day 0. Depending on when the trial was joined, patients receive cyclophosphamide IV over 3 hours or bendamustine IV over 30-60 minutes or cyclophosphamide IV over 3 hours and bendamustine IV over 30-60 minutes on day 3. Patients also receive bendamustine IV over 30-60 minutes on day 4. Beginning day 5, patients receive tacrolimus IV followed by orally (PO) once daily (QD) or twice daily (BID) for 6 months and mycophenolate mofetil PO thrice daily (TID) until day 100. Beginning day 7, patients receive filgrastim-sndz subcutaneously (SC) QD until blood cell levels return to normal. SCHEDULE II (LYMPHOID MALIGNANCIES): Patients receive fludarabine IV over 1 hour, bendamustine IV over 30-60 minutes on days -5 to -3 and undergo TBI on day -1 and stem cell transplantation over 2-6 hours on day 0. Depending on when the trial was joined, patients receive cyclophosphamide IV over 3 hours or bendamustine IV over 30-60 minutes or cyclophosphamide IV over 3 hours and bendamustine IV over 30-60 minutes on day 3. Patients also receive bendamustine IV over 30-60 minutes on day 4. Beginning day 5, patients receive tacrolimus IV followed by PO QD or BID for 6 months and mycophenolate mofetil PO TID until day 100. Beginning day 7, patients receive filgrastim-sndz SC QD until blood cell levels return to normal. CD20+ patients receive rituximab IV over 4-6 hours on days -13, -6, 1, and 8. After completion of study treatment, patients are followed weekly for 3 months, every 3 months in year 1, and every 6 months in year 2.