A Phase II Study of Allo-HCT for B-Cell NHL Using Zevalin, Fludarabine and Melphalan

Inclusion Criteria: * 6/6/human leukocyte antigen (HLA) matched sibling donor or related donor, or acceptable matched unrelated donor * Biopsy (Bx) proven diagnosis of LG (including small lymphocytic lymphoma \[SLL\]/chronic lymphocytic leukemia \[CLL\], lymphoplasmacytic lymphoma, marginal zone, mucosa-associated lymphoid tissue \[MALT\] lymphoma and follicular lymphoma \[FL\] grade 1 and 2), IG (FL grade 3 and DLCL) or MCL NHL * Prior demonstrated monoclonal CD20+ malignant B-Cell population in lymph nodes and/or BM Bx specimen * LG NHL; must have relapsed after achieving a complete response (CR) or partial response (PR) to prior therapy or have never responded to prior therapy, including chemotherapy and/or MAb therapy * MCL NHL in any disease state * Other aggressive B-cell lymphomas (excluding Burkitt lymphoma or Burkitt-like lymphoma) having had at least one relapse or having been refractory to chemotherapy * Bone marrow (BM) aspiration and Bx ( =\< 42 days prior to imaging dose) which show \< 25% lymphomatous involvement of total cellularity; in CLL, peripheral lymphocyte count \< 5000/mm\^3 * Salvage chemotherapy/MAbs to reduce BM lymphomatous involvement and reduce disease bulk allowed * Normal renal function test with serum creatinine of =\< 1.5 mg/dl, or a creatinine clearance of \>= 60 ml/min * Adequate pulmonary function as measured by forced expiratory volume in one second (FEV1) \> 65% of predicted measured, or a diffusing capacity of carbon monoxide (DLCO) \>= 50% of predicted measured * Cardiac Ejection fraction of \> 50% by Echocardiogram (ECHO) or multi gated acquisition (MUGA) * Adequate liver function tests with a bilirubin of =\< 1.5 x normal and serum glutamic oxaloacetic transaminase (SGOT) or serum glutamic pyruvic transaminase (SGPT) =\< 2 x normal * Negative Human Immunodeficiency Virus (HIV) antibody * Karnofsky Performance Status (KPS) \> 80 * No active Central Nervous System (CNS) disease or prior history of CNS disease * Involved field External Beam Therapy (EBT) to area excluding lung, heat, liver, and kidney allowed, but evaluated on a case-by-case basis * Recovery from last therapy and therapy dose (Y-90 Zevalin) must be \>= 4 weeks from prior systemic chemotherapy * DONOR: Age \< 75 years * DONOR: HLA genotypically identical related donor or acceptable matched unrelated donor * DONOR: Must consent to G-CSF administration and leukapheresis for matched sibling donor, but for unrelated donor, the donor will sign a standard consent for donation at their designated donor or collection center * DONOR: Must have adequate veins for leukapheresis or agree to placement of a Central Venous Catheter (CVC \[femoral, subclavian\]) Exclusion Criteria: * Presence of Human Anti-Zevalin Antibody (HAZA) * Prior radioimmunotherapy (RIT) * Prior AHSCT; but prior aHSCT is allowed; prior fractionated total body irradiation (FTBI) in the conditioning regimen will be evaluated on an individual basis * Prior malignancy, except for: adequately treated basal cell or squamous cell skin cancer; adequately treated noninvasive carcinomas; or other cancer from which the patient has been disease-free for at least 5 years; myelodysplastic syndromes (MDS) is excluded from this criterion * Active evidence of Hepatitis B or C infection; hepatitis B surface antigen positive * Total peripheral lymphocyte count \> 5,000/mm\^3 if SLL/CLL * Burkitt lymphoma or Burkitt-like lymphoma * DONOR: Age \< 12 years * DONOR: Identical twin * DONOR: Pregnancy * DONOR: HIV infection * DONOR: Inability to achieve adequate venous access * DONOR: Known allergy to G-CSF * DONOR: Current serious systemic illness or any disease that may preclude the use of G-CSF (eg, recent thromboembolic event); for unrelated donors, considered ineligible by National Marrow Donor Program (NMDP) donor evaluation center