Gemcitabine Hydrochloride, Clofarabine, and Busulfan Before Donor Stem Cell Transplant in Treating Patients With Refractory B-Cell or T-Cell Non-Hodgkin Lymphoma or Hodgkin Lymphoma

Inclusion Criteria: * Patients with refractory B-cell or T-cell non-Hodgkin's lymphoma or Hodgkin's lymphoma who are eligible for allogeneic transplantation * An 8/8 human leukocyte antigen (HLA) matched (high resolution typing at A, B, C, DRB1) sibling or unrelated donor * Left ventricular ejection fraction (EF) \>= 45% * Forced expiratory volume in one second (FEV1) \>= 50% * Forced vital capacity (FVC) \>= 50% * Diffusing capacity of the lung for carbon monoxide (DLCO) \>= 50% * Estimated serum creatinine clearance \>= 50 ml/min (using the Cockcroft-Gault formula) * Serum creatinine =\< 1.6 mg/dL * Serum bilirubin =\< 2 x upper limit of normal * Serum glutamate pyruvate transaminase (SGPT) =\< 2 x upper limit of normal * Voluntary signed 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performance of any study-related procedure not part of normal medical care, with the understanding that consent may be withdrawn by the subject at any time without prejudice to future medical care * Men and women of reproductive potential must agree to follow accepted birth control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 …