This randomized phase II trial includes a blood stem cell transplant from an unrelated donor to treat blood cancer. The treatment also includes chemotherapy drugs, but in lower doses than conventional (standard) stem cell transplants. The researchers will compare two different drug combinations used to reduce the risk of a common but serious complication called "graft versus host disease" (GVHD) following the transplant. Two drugs, cyclosporine (CSP) and sirolimus (SIR), will be combined with either mycophenolate mofetil (MMF) or post-transplant cyclophosphamide (PTCy). This part of the transplant procedure is the main research focus of the study.

Ages Eligible for Study: Child, Adult, Senior

Sexes Eligible for Study: All

- Ages > 50 years with hematologic malignancies treatable by unrelated hematopoietic cell transplant (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 hematopoietic cell transplant-comorbidity index (HCT-CI) score of > 3; 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 with chronic lymphocytic leukemia (CLL)

- Ages =< 50 years 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 nonHodgkin 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 1st CR; or 4) 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; 5) patients failing to achieve a response to ibrutinib as first-line therapy; 6) patients not responding to ibrutinib, idelalisib, or venetoclax as salvage therapy or intolerant of these agents as salvage therapy due to side effects; all CLL patients must have received prior myelosuppressive chemotherapy

- 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 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)/chronic myelomonocytic leukemia (CMML) – patients must have < 5% marrow blasts at time of transplant

- Waldenstrom’s macroglobulinemia – must have failed 2 courses of therapy

- HLA-MATCHED UNRELATED DONOR: 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

- HLA-MATCHED UNRELATED 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

- HLA-MATCHED UNRELATED 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

- HLA-MATCHED UNRELATED DONOR: Only granulocyte colony-stimulating factor (G-CSF) mobilized PBSC will be permitted as a HSC source on this protocol

- HLA-MISMATCHED UNRELATED DONOR: 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

-- * HLA class I HLA-A, -B, -C allele matched donors allowing for any one or two DRB1 and/or DQB1 antigen/allele mismatch

- HLA-MISMATCHED UNRELATED DONOR: HLA-matching must be based on results of high resolution typing at HLA-A, –B, -C, - DRB1, and –DQ

- HLA-MISMATCHED UNRELATED 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

Other eligibility criteria may apply.

- Patients with rapidly progressive intermediate or high grade NHL

- Patients with a diagnosis of chronic myelomonocytic leukemia (CMML) who have not received induction chemotherapy

- Patients with refractory anemia with excess blasts (RAEB) who have not received myelosuppressive chemotherapy i.e. induction chemotherapy and will receive conditioning Regimen C (fludarabine and total body irradiation [TBI]) will be excluded; patients with RAEB who have not received myelosuppressive chemotherapy but who will receive conditioning Regimen A or B are eligible for this study as long as other inclusion and exclusion criteria are met

- CNS involvement with disease refractory to intrathecal chemotherapy

- Presence of circulating blasts (in the blood) detected by standard pathology for patients with AML, ALL or CML

- Presence of >= 5% circulating leukemic blasts (in the blood) detected by standard pathology for patients with MDS/MPS/CMML

- Fertile men or women unwilling to use contraceptive techniques during and for 12 months following treatment

- Females who are pregnant or breast-feeding

- Patients with active non-hematological malignancies (except non-melanoma skin cancers) or those with non-hematological 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

- Organ dysfunction

-- * Cardiac ejection fraction < 35% (or, if unable to obtain ejection fraction, shortening fraction of < 26%); ejection fraction is required if age > 50 years or there is a history of anthracycline exposure or history of cardiac disease; patients with a shortening fraction < 26% may be enrolled if approved by a cardiologist

-- * Pulmonary:

--- ** Diffusing capacity of the lungs for carbon monoxide (DLCO) < 40%, total lung capacity (TLC) < 40%, forced expiratory volume in the first second of breath (FEV1) < 40% and/or receiving supplementary continuous oxygen

--- ** The FHCRC PI of the study must approve of enrollment of all patients with pulmonary nodules

- Liver function abnormalities: 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

- Karnofsky scores < 60 or Lansky score < 50

- Patient has poorly controlled hypertension and on multiple antihypertensives

- Human immunodeficiency virus (HIV) positive patients

- Active bacterial or fungal infections unresponsive to medical therapy

- 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 peripheral blood stem cell (PBSC)

- DONOR: Patients who are homozygous at the mismatched HLA class I 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

Other exclusion criteria may apply.