Sirolimus, Cyclosporine, and Mycophenolate Mofetil in Preventing Graft-versus-Host Disease in Treating Patients With Blood Cancer Undergoing Donor Peripheral Blood Stem Cell Transplant

Inclusion Criteria: * Ages \> 50 years with hematologic malignancies treatable by related or 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 high dose transplant (\> 40% risk of transplant-related mortality \[TRM\]); this criterion can include patients with a HCT-comorbidity index (CI) score of \>= 1; transplants should be approved for these inclusion criteria by the principal investigators at the collaborating centers and at Fred Hutchinson Cancer Research Center (FHCRC); all children \< 12 years must be discussed with the FHCRC principal investigator (PI) prior to registration * Ages =\< 50 years of age with chronic lymphocytic leukemia (CLL) * Ages =\< 50 years of age with hematologic diseases treatable by allogeneic HCT who refuse a high-dose HCT; transplants must be approved for these inclusion criteria by the principal investigators at the collaborating centers and at FHCRC * Aggressive non-Hodgkin lymphomas (NHL) and other histologies such as diffuse large B cell NHL: not eligible for autologous HCT, not eligible for high-dose allogeneic HCT, or after failed autologous HCT * Mantle cell NHL: may be treated in first complete remission (CR); (diagnostic lumbar puncture \[LP\] required pre-transplant) * Low grade NHL: with \< 6 month duration of CR between courses of conventional therapy * CLL: must have either 1) failed to meet National Cancer Institute (NCI) Working Group criteria for complete or partial response after therapy with a regimen containing fludarabine (FLU) (or another nucleoside analog, e.g. cladribine \[2-CDA\], pentostatin) or experience disease relapse within 12 months after completing therapy with a regimen containing FLU (or another nucleoside analog); 2) failed FLU-cyclophosphamide (CY)-Rituximab (FCR) combination chemotherapy at any time point; or 3) have "17p deletion" cytogenetic abnormality; patients should have received induction chemotherapy but could be transplanted in first (1st) CR; 4) patients with a diagnosis of CLL (or small lymphocytic lymphoma) that progresses to prolymphocytic leukemia (PLL); or 5) patients with T-cell CLL or PLL * Hodgkin lymphoma: must have received and failed frontline therapy * Multiple myeloma: 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 myeloid leukemia (CML): patients in chronic phase 1 (CP1) must have failed or be intolerant of tyrosine kinase inhibitors (TKIs); patients beyond CP1 will be accepted if they have \< 5% marrow blasts at time of transplant * Myelodysplasia (MDS)/myeloproliferative syndrome (MPS): patients must have \< 5% marrow blasts at time of transplant * Waldenstrom's macroglobulinemia: must have failed 2 courses of therapy * Patients with related or unrelated donors for whom the best available donor is: a) mismatched at antigen level for any single class I locus (HLA-A, -B, -C) +/- an additional class I mismatch at the allele level OR mismatched at the allele level for any 2 class I loci (if typed at the molecular level) OR mismatched at the antigen or allele level for class II loci HLA-DRB1 and/or - DQB1; must be matched for at least one DRB1 allele and one DQB1 allele; b) there is a likelihood of rapid disease progression while HLA typing and results of a preliminary search and the donor pool suggests that a 10/10 HLA-A, B, C, DRB1 and DQB1 matched donor will not be found; c) there is no HLA-A, -B or -C one locus allelic mismatched donor available * DONOR: Related or unrelated volunteer donors who are mismatched with the recipient within one of the following limitations: * Mismatch for one HLA class I antigen with or without an additional mismatch for one HLA-class I allele, but matched for HLA-DRB1 and HLA-DQ, OR * Mismatched for two HLA class I alleles, but matched for HLA-DRB1 and HLA-DQ, OR * HLA class I HLA-A, -B, -C allele matched donors allowing for any one or two DRB1 and/or DQB1 antigen allele mismatch * DONOR: HLA-matching must be based on results of high resolution typing at HLA-A, -B, -C, -DRB1, and -DQB * DONOR: If the patient is homozygous at the mismatch HLA class I locus or II locus, the donor must be heterozygous at that locus and one allele must match the patient (i.e., patient is homozygous A\*01:01 and donor is heterozygous A\*01:01, A\*02:01); this mismatch will be considered a one-antigen mismatch for rejection only * 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 donor should be excluded if any of the flow cytometric B and T cell cytotoxic cross match assays are positive * DONOR: Only filgrastim (G-CSF) mobilized PBSC only will be permitted as a hematopoietic stem cell (HSC) source on this protocol Exclusion Criteria: * Patients for whom the best available donor is mismatched at both HLA class I and class II * A positive cross-match exists between the donor and recipient * Patients with rapidly progressive intermediate or high grade NHL * Patients with a diagnosis of chronic myelomonocytic leukemia (CMML) * Patients with refractory anemia with excess blasts (RAEB)-2 who have not received myelosuppressive chemotherapy i.e. induction chemotherapy * Presence of circulating leukemic blasts (in the peripheral blood) detected by standard pathology for patients with AML, ALL or CML * Presence of \>= 5% circulating leukemic blasts (in the peripheral blood) detected by standard pathology for patients with MDS/MPS * Central nervous system (CNS) involvement with disease refractory to intrathecal chemotherapy * Fertile men or women unwilling to use contraceptives during and for up to 12 months following treatment * Female patients who are pregnant or breast-feeding * Human immunodeficiency virus (HIV) positive patients * Patients with active non-hematologic malignancies (except non-melanoma skin cancers) or those with non-hematologic malignancies (except non-melanoma skin cancers) who have been rendered with no evidence of disease, but have a greater than 20% chance of having disease recurrence within 5 years; this exclusion does not apply to patients with non-hematologic malignancies that do not require therapy * Fungal infections with radiological progression after receipt of amphotericin B or active triazole for greater than 1 month * Patients with active bacterial or fungal infections unresponsive to medical therapy * Cardiac ejection fraction \< 35% (or, if unable to obtain ejection fraction, shortening fraction of \< 26%); ejection fraction is required if the patient is \> 50 years of age, or history of cardiac disease or anthracycline exposure; patients with a shortening fraction \< 26% may be enrolled if approved by a cardiologist * Corrected diffusion capacity of carbon monoxide (DLCO) \< 40%, total lung capacity (TLC) \< 40%, forced expiratory volume in one second (FEV1) \< 40% and/or receiving supplementary continuous oxygen; if unable to perform complete pulmonary function tests (PFTs), patients will be excluded if their oxygen saturation is \< 95% with a formal six-minute walk test (ambulatory oximetry) * 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, or symptomatic biliary disease * Patients with poorly controlled hypertension on multiple antihypertensives * Karnofsky scores \< 60 or Lansky score \< 50 * All patients receiving antifungal therapy voriconazole, posaconazole, or fluconazole must have sirolimus reduced according to the Standard Practice Antifungal Therapy Guidelines * The addition of cytotoxic agents for "cytoreduction" with the exception of tyrosine kinase inhibitors (such as imatinib), cytokine therapy, hydroxyurea, low dose cytarabine, chlorambucil, or Rituxan will not be allowed within three weeks of the initiation of conditioning * DONOR: Donor (or centers) who will exclusively donate marrow * DONOR: Donors who are HIV-positive and/or, medical conditions that would result in increased risk for G-CSF mobilization and harvest of PBSC * DONOR: Patients who are homozygous at the mismatched HLA class I locus or II locus, the donor is excluded if homozygous at the mismatched locus (i.e., patient is homozygous A\*01:01 and donor is homozygous A\*02:01); this type of mismatch is considered a two-antigen mismatch and is not allowed