Prolonged Mycophenolate Mofetil and Truncated Cyclosporine Postgrafting Immunosuppression to Reduce Life-Threatening GVHD After Unrelated Donor Peripheral Blood Cell Transplantation Using Nonmyeloablative Conditioning for Patients With Hematologic Malignancies and Renal Cell Carcinoma - A Multi-Center Trial

Inclusion Criteria: * Ages \> 50 years with hematologic malignancies treatable by unrelated HCT * Ages =\< 50 years of age with hematologic diseases treatable by allogeneic HCT who through pre-existing medical conditions or prior therapy are considered to be at high risk for regimen related toxicity associated with a conventional transplant (\> 40% risk of transplant-related mortality \[TRM\]) or those patients who refuse a conventional HCT; transplants must be approved for these inclusion criteria by both the participating institution's patient review committee such as the Patient Care Conference (PCC at the Fred Hutchinson Cancer Research Center \[FHCRC\]) and by the principal investigator at the collaborating center; patients =\< 50 years of age who have received previous high-dose transplantation do not require patient review committee approval; all children \< 12 years must be discussed with the FHCRC primary investigator (PI) prior to registration * Patients with metastatic renal cell carcinoma with the histologic subtypes of clear cell, papillary and medullary may be accepted regardless of age * The following diseases will be permitted although other diagnoses can be considered if approved by PCC or the participating institution's patient review committees and the principal investigator: * Aggressive non-Hodgkin lymphomas (NHLs) and other histologies such as diffuse large B cell NHL-not eligible for autologous hematopoietic stem cell transplant (HSCT), not eligible for conventional myeloablative HSCT, or after failed autologous HSCT * Low grade NHL- with \< 6 month duration of complete remission (CR) between courses of conventional therapy * Mantle cell NHL-may be treated in first CR * Chronic lymphocytic leukemia (CLL)- Must be refractory to fludarabine; patients who fail to have a complete or partial response after therapy with a regimen containing fludarabine (or another nucleoside analog, e.g. 2-cladribine \[CDA\], pentostatin) or experience disease relapse within 12 months after completing therapy with a regimen containing fludarabine (or another nucleoside analog) * Hodgkin disease (HD)- must have received and failed frontline therapy * Multiple myeloma (MM)- must have received prior chemotherapy; consolidation of chemotherapy by autografting prior to nonmyeloablative HCT is permitted * Acute myeloid leukemia (AML)- must have \< 5% marrow blasts at the time of transplant. * Acute lymphocytic leukemia (ALL)- must have \< 5% marrow blasts at the time of transplant * Chronic myelogenous leukemia (CML)- Patients will be accepted in chronic phase or accelerated phase; patients who have received prior autografts after high dose therapy or have undergone intensive chemotherapy with filgrastim (G-CSF)-mobilized peripheral blood mononuclear cells (G-PBMC) autologous or conventional HCT for advanced CML may be enrolled provided they are in CR or CP and have \< 5% marrow blasts at time of transplant * Myelodysplastic syndrome (MDS)/myeloproliferative disorder (MPD)- Only patients with MDS/refractory anemia (RA) or MDS/refractory anemia with ringed sideroblasts (RARS) will be eligible for this protocol; additionally patients with myeloproliferative syndromes (MPS) will be eligible; those patients with MDS or MPS with \> 5% marrow blasts (including those with transformation to AML) must receive cytotoxic chemotherapy and achieve \< 5% marrow blasts at time of transplant * Renal cell carcinoma- Must have evidence of disease not amenable to surgical cure or history of or active metastatic disease by radiological and histologic criteria * DONOR: FHCRC matching allowed will be grade 1.0 to 2.1: Unrelated donors who are prospectively: * Matched for human leukocyte antigen (HLA)-A, B, C, major histocompatibility complex, class II, DR beta 1 (DRB1) and major histocompatibility complex, class II, DQ beta 1 (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: 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\*0201, and this type of mismatch is not allowed * DONOR: G-PBMC only will be permitted as a HSC source on this protocol Exclusion Criteria: * Patients with rapidly progressive intermediate or high grade NHL * Renal cell carcinoma patients * With expected survival of less than 6 months * Disease resulting in severely limited performance status (\< 70%) * Any vertebral instability * History of brain metastases * Central nervous system (CNS) involvement with disease refractory to intrathecal chemotherapy * Fertile men or women unwilling to use contraceptive techniques during and for 12 months following treatment * Females who are pregnant * Patients with non-hematological tumors except renal cell carcinoma * Fungal infections with radiological progression after receipt of amphotericin B or active triazole for greater than 1 month * Cardiac ejection fraction \< 35%; ejection fraction is required if there is a history of anthracycline exposure or history of cardiac disease * Diffusing capacity of the lung for carbon monoxide (DLCO) \< 40% and/or receiving supplementary continuous oxygen * The FHCRC PI of the study must approve of enrollment of all patients with pulmonary nodules * Patients with clinical or laboratory evidence of liver disease would be evaluated for the cause of liver disease, its clinical severity in terms of liver function, and the degree of portal hypertension; patients will be excluded if they are found to have fulminant liver failure, cirrhosis of the liver with evidence of portal hypertension, alcoholic hepatitis, esophageal varices, a history of bleeding esophageal varices, hepatic encephalopathy, uncorrectable hepatic synthetic dysfunction evinced by prolongation of the prothrombin time, ascites related to portal hypertension, bridging fibrosis, bacterial or fungal liver abscess, biliary obstruction, chronic viral hepatitis with total serum bilirubin \> 3 mg/dL, and symptomatic biliary disease * Karnofsky scores \< 60 (except renal cell carcinoma \[RCC\]) * Patients with \> grade II hypertension by Common Toxicity Criteria (CTC) * Human immunodeficiency virus (HIV) positive patients * The addition of cytotoxic agents for "cytoreduction" with the exception of hydroxyurea and imatinib mesylate will not be allowed within two weeks of the initiation of conditioning * DONOR: Marrow donors * DONOR: Donors who are HIV-positive and/or, medical conditions that would result in increased risk for G-CSF mobilization and harvest of G-PBMC