Autologous Stem Cell Transplant Followed by Donor Stem Cell Transplant in Treating Patients With Relapsed or Refractory Lymphoma

Inclusion Criteria: * Patients with lymphoma (non-Hodgkin lymphoma \[NHL\], chronic lymphocytic leukemia/small lymphocytic lymphoma \[CLL/SLL\] or Hodgkin's lymphoma) with primary refractory or relapsed disease after standard chemotherapy at high risk of relapse with conventional autografting; patients with a diagnosis of CLL (or small lymphocytic lymphoma) or diagnosis of CLL that progresses to prolymphocytic leukemia (PLL), or T-cell CLL or PLL * Must have an HLA genotypically or phenotypically identical related donor or, at a minimum, a high likelihood of identifying an HLA-matched unrelated donor; the determination of availability of a suitable unrelated donor may be based on a World-Book search * Cross-over to other tandem autologous-allogeneic research protocol (#2241) will be allowed if the patient loses the suitable HLA-matched related or unrelated donor but has an available HLA-haploidentical donor before receiving the allogeneic transplantation and if the patient meets the eligibility criteria of the subsequent study * Cross-over from other tandem autologous-allogeneic research protocol (#2241) will be allowed if a suitable HLA-matched related or unrelated donor is identified before receiving the allogeneic transplantation and if the patient meets the eligibility criteria of the subsequent study * Signed informed consent * Detectable tumor on radiographic studies or bone marrow biopsy prior to mobilization regimen * Expected survival \>= 3 months from study entry * DONOR: HLA genotypically or phenotypically identical related donor * DONOR: Must consent to granulocyte-colony stimulating factor (G-CSF) (filgrastim) administration and leukapheresis for both PBSC allograft and subsequent donor lymphocyte infusion (DLI) * DONOR: Must have adequate veins for leukapheresis or agree to placement of central venous catheter (femoral or subclavian) * DONOR: Age \< 75 years (yrs), older donors may be considered after review at Patient Care Conference * DONOR: Fred Hutchinson Cancer Research Center (FHCRC) matching allowed will be grades 1.0 to 2.1; unrelated donors who are prospectively: matched for HLA-A, B, C, DRB1 and DQB1 by high resolution typing; only a single allele disparity will be allowed for HLA-A, B, or C as defined by high resolution typing * DONOR: Donors are excluded when preexisting immunoreactivity is identified that would jeopardize donor hematopoietic cell engraftment; this determination is based on the standard practice of the individual institution; the recommended procedure for patients with 10 of 10 HLA allele level (phenotypic) match is to obtain a panel reactive antibody (PRA) screens to class I and class II antigens for all patients before HCT; if the PRA shows \> 10% activity, then flow cytometric or B and T cell cytotoxic cross matches should be obtained; the donor should be excluded if any of the cytotoxic cross match assays are positive; for those patients with an HLA class I allele mismatch, flow cytometric or B and T cell cytotoxic cross matches should be obtained regardless of the PRA results; a positive anti-donor cytotoxic crossmatch is an absolute donor exclusion * DONOR: Patient and donor pairs homozygous at a mismatched allele in the graft rejection vector are considered a two-allele mismatch, i.e., the patient is A\*0101 and the donor is A\*0102, and this type of mismatch is not allowed * DONOR: Only G-CSF mobilized peripheral blood mononuclear cells (PBMC) only will be permitted as a hematopoietic stem cell (HSC) source on this protocol Exclusion Criteria: * Life expectancy severely limited by disease other than lymphoma * Prior autologous hematopoietic stem cell transplant * Patients at high risk of veno-occlusive disease of the liver (criteria not yet rigorously defined but includes bilirubin \> 2.0 mg and serum glutamic oxaloacetic transaminase \[SGOT\] or serum glutamate pyruvate transaminase \[SGPT\] \> 2 x normal); patients may be accepted outside of this range if cleared by gastrointestinal (GI) consult * Cardiac ejection fraction (EF) \< 40% on multi-gated acquisition (MUGA) scan or cardiac echocardiogram (echo) (or if unable to obtain ejection fraction, shortening fraction of \< 26%); patients with active or a history of cardiac disease should be evaluated with appropriate cardiac studies and/or consult; ejection fraction is required if age \> 50 years or there is a history of anthracyclines or history of cardiac disease; patients with a shortening fraction \< 26% may be enrolled if approved by a cardiologist * Baseline serum-creatinine \> 2.0 mg/dl and a calculated or measured creatinine clearance of \< 50 ml/minute * Seropositive for the human immunodeficiency virus (HIV) * Pulmonary dysfunction as measured by a corrected diffusing capacity of the lung for carbon monoxide (DLCO) \< 50% of predicted total lung capacity (TLC) \< 30%, forced expiratory volume in 1 second (FEV1) \< 30% and/or receiving supplementary continuous oxygen; the FHCRC principal investigator (PI) of the study must approve enrollment of all patients with pulmonary nodules * Pregnancy or breast-feeding * Patients with poorly controlled hypertension despite hypertensive medication * Karnofsky score less than 60; pediatric criteria: Lansky Play-Performance Score \< 40 * Patients with cluster of differentiation (CD)34 selected auto grafts * Patients with active non-hematologic malignancies (except non-melanoma skin cancers); this exclusion does not apply to patients with non-hematologic malignancies that do not require therapy * Patients with a history of non-hematologic malignancies (except non-melanoma skin cancers) currently in a complete remission, who are less than 5 years from the time of complete remission, and have a \> 20% risk of disease recurrence; this exclusion does not apply to patients with non-hematologic malignancies that do not require therapy * DONOR: Identical twin * DONOR: Age less than 12 years * DONOR: Pregnancy * DONOR: Human immunodeficiency virus (HIV) seropositivity * DONOR: Inability to achieve adequate venous access * DONOR: Known allergy to G-CSF * DONOR: Current serious systemic illness * DONOR: Failure to meet FHCRC criteria for stem cell donation * DONOR: Donor (or centers) who will exclusively donate marrow