Brief Summary: This phase I trial studies the best dose and how well bendamustine works with standard chemotherapy (fludarabine, rituximab) in treating participants with lymphoid cancers undergoing stem cell transplant. Drugs used in chemotherapy, such as fludarabine, bendamustine, and rituximab, 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 before a stem cell transplant helps stop the growth of cancer cells in the bone marrow, including normal blood-forming cells (stem cells) and cancer cells. When the healthy stem cells from a donor are infused into the participant, they may help the participant's bone marrow make stem cells, 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. Giving rituximab and methotrexate after the transplant may stop this from happening.

Detailed Description: PRIMARY OBJECTIVES: I. To determine engraftment and dose limiting toxicity (DLT) of bendamustine in patients with lymphoid malignancies undergoing non-myeloablative allogeneic hematopoietic transplantation. SECONDARY OBJECTIVES: I. To monitor the risk of graft-versus-host disease (GVHD) and clinical responses. OUTLINE: This is a dose escalation study of bendamustine. Participants receive rituximab intravenously (IV) over 5-7 hours on days -13 and -6, fludarabine IV over 1 hour and bendamustine IV over 1 hour on days -5 to -3, and tacrolimus IV starting on day -2 and orally (PO) after hospital discharge for 6 to 8 months. Participants with matched unrelated donor (MUD) receive thymoglobulin on days -2 and -1. Participants undergo allogenic stem cell transplant over 30-45 minutes on day 0. Participants receive rituximab IV over 5-7 hours on days 1 and 8 and methotrexate IV over 30 minutes on days 1, 3, and 6. Participants with MUD also receive methotrexate IV on day 11. Participants receive filgrastim (G-CSF) subcutaneously (SC) once daily starting on day 7 until white blood cell counts recover. After completion of study treatment, participants are followed up every 3 months during year 1 and every 6 months for up to 3 years.