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performance of any study-related procedure not part of normal medical care, with the understanding that consent may be withdrawn by the subject at any time without prejudice to future medical care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 the study

Exclusion Criteria:

* Patient with active central nervous system (CNS) disease
* Pregnancy (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
* Active hepatitis B, either active carrier (hepatitis B virus surface antigen \[HBsAg\] +) or viremic (hepatitis B virus \[HBV\] deoxyribonucleic acid \[DNA\] \>= 10,000 copies/mL, or \>= 2,000 IU/mL)
* Evidence of either cirrhosis or stage 3-4 liver fibrosis in patients with chronic hepatitis C or positive hepatitis C serology
* Human immunodeficiency virus (HIV) infection
* Active uncontrolled bacterial, viral or fungal infections
* Exposure to other investigational drugs within 2 weeks before enrollment
* Grade \>= 3 non-hematologic toxicity from previous therapy that has not resolved to =\< grade 1
* Radiation therapy to head and neck (excluding eyes), and internal organs of chest, abdomen or pelvis in the month prior to enrollment
* Prior whole brain irradiation
* Prior autologous stem-cell transplant (SCT) in the prior 3 months