Fludarabine, Bendamustine, and Rituximab in Treating Participants With Lymphoid Cancers Undergoing Stem Cell Transplant

Inclusion Criteria: * Patients with CD20 + chronic lymphocytic leukemia (CLL), marginal zone, mantle cell and follicular lymphoma or T-cell lymphoid malignancies who are eligible for allogeneic transplantation. * Patients with relapsed diffuse large B-cell lymphoma may be included if they were not eligible for autologous transplantation. * A fully-matched sibling donor or matched unrelated donor. * Left ventricular ejection fraction (EF) \> 40% with no uncontrolled arrhythmias or symptomatic heart disease. * Forced expiratory volume in one second (FEV1), forced vital capacity (FVC) and diffusion capacity of the lung for carbon monoxide (DLCO) \> 40%. * Serum creatinine \< 1.6 mg/dL. * Serum bilirubin \< 3 X upper limit of normal. * Serum glutamate pyruvate transaminase (SGPT) \< 3 X upper limit of normal. * Voluntary signed, written Institutional Review Board (IRB)-approved informed consent before performance of any study-related procedure not part of normal medical care, with the understanding that consent may be withdrawn by the subject at any time without prejudice to future medical care. * Men and women of reproductive potential must agree to follow accepted birth control methods for the duration of the study. Female subject is either post-menopausal or surgically sterilized or willing to use an acceptable method of birth control (i.e., a hormonal contraceptive, intra-uterine device, diaphragm with spermicide, condom with spermicide, or abstinence) for the duration of the study. Male subject agrees to use an acceptable method for contraception for the duration of the study. Exclusion Criteria: * Patient with active central nervous system (CNS) disease. * Pregnant (positive beta human chorionic gonadotropin \[HCG\] test in a woman with child bearing potential defined as not post-menopausal for 12 months or no previous surgical sterilization) or currently breast-feeding. Pregnancy testing is not required for post-menopausal or surgically sterilized women. * Known infection with human immunodeficiency virus (HIV), human T-lymphotropic virus (HTLV)-I, hepatitis B, or hepatitis C. * Patients with other malignancies diagnosed within 2 years prior to study day-13 (except skin squamous or basal cell carcinoma). * Active uncontrolled bacterial, viral or fungal infections. * History of stroke within 6 months. * Myocardial infarction within the past 6 months prior to study day 1, or has New York Heart Association (NYHA) class III or IV heart failure or arrhythmias, unstable angina, uncontrolled congestive heart failure or arrhythmias, or electrocardiographic evidence of acute ischemia or active conduction system abnormalities. Prior to study entry, any electrocardiography (ECG) abnormality at screening must be documented by investigator as not medically relevant. * A prior allogeneic transplant. * Serious medical or psychiatric illness likely to interfere with participation in this clinical study. * Patient has received other investigational drugs within 3 weeks before enrollment. * Hypersensitivity to bendamustine. * Prior known refractoriness to bendamustine.