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 thera…