The Lowest Effective Dose of Post-Transplantation Cyclophosphamide in Combination With Sirolimus and Mycophenolate Mofetil as Graft-Versus-Host Disease Prophylaxis After Reduced Intensity Conditioning and Peripheral Blood Stem Cell Transplantation

* INCLUSION CRITERIA: Recipient * Participants must have a histologically or cytologically confirmed hematologic malignancy with standard indication for allogeneic hematopoietic cell transplantation limited to one of the following: * Acute myeloid leukemia (AML) of intermediate or adverse risk disease by the 2017 European LeukemiaNet criteria in first morphologic complete remission (\<5% blasts in the bone marrow, no detectable abnormal peripheral blasts, and no extramedullary disease) * AML of any risk in second or subsequent morphologic complete remission * Acute lymphoblastic leukemia in first or subsequent complete remission * Myelodysplastic syndrome of intermediate or higher score by the Revised International Prognostic Scoring System (IPSS-R) * Primary myelofibrosis of intermediate-2 or higher risk by the DIPSS * Chronic myelomonocytic leukemia * Chronic myelogenous leukemia resistant to or intolerant of \>= 3 tyrosine kinase inhibitors or with history of accelerated phase or blast crisis * B-cell lymphoma including Hodgkin lymphoma that has relapsed within 1 year of completion of primary treatment, relapsed after autologous transplantation, or has progressed through at least 2 lines of therapy * Chronic lymphocytic leukemia with 17p deletion and/or unmutated IgHV or refractory to or intolerant of both BTK and PI3K inhibitors * Mature T or NK neoplasms as defined in the WHO guidelines of sufficient type and severity for allogeneic HCT based on the Prognostic Index for T-cell lymphoma (PIT) score of low-intermediate risk or higher or on recently published clinical practice guidelines * Hematologic malignancy of dendritic cell or histiocytic cell type * Multiple myeloma, stage III, relapsing after therapy with both a proteasome inhibitor and an immunomodulatory drug (IMiD) * Age \>= 50 years or age 18-49 years and also meeting one of the following criteria: * Prior myeloablative HCT * Prior exposure to inotuzumab, gemtuzumab, or other agent that increases the risk for sinusoidal obstruction syndrome. * Hematopoietic Cell Transplantation- Comorbidity Index (HCT-CI) \>= 3 * Karnofsky performance score \<80 * Co-morbidity considered by the treating physician to be exclusionary of myeloablative conditioning * At least one potentially suitable HLA-haploidentical or 10/10 (HLA-A, B, C, DR, DQ) related or unrelated donor for HCT * Karnofsky performance score \>= 70 * Adequate organ function defined as possessing all of the following: * Cardiac ejection fraction \>= 45% by 2D ECHO; * Forced expiratory volume-1 (FEV-1), forced vital capacity (FVC), and diffusing capacity of the lung for carbon monoxide (DLCO) (corrected for hemoglobin) all of \>= 50% predicted; * Estimated serum creatinine clearance of \>= 60 ml/minute/1.73m\^2 calculated using eGFR in the clinical lab; * Total bilirubin \<= 2X the upper limit of normal; * Alanine aminotransferase and aspartate aminotransferase \<= 3X the upper limit of normal. * Individuals of child-bearing potential (IOCBP) and participants who can father children must agree to use highly effective contraception (hormonal or barrier method of birth control; abstinence) prior to study entry and for at least one year post-transplant. * IOCBP must have a negative serum or urine pregnancy test within 7 days prior to initiation of conditioning regimen. * Ability of participant to understand and the willingness to sign a written informed consent document. EXCLUSION CRITERIA: Recipient * Participants who are receiving any other investigational agents. Prior experimental therapies must have been completed at least 2 weeks prior to the date of beginning conditioning. * Active nursing. * Active malignancy of non-hematopoietic type (excluding non-melanoma skin cancers) which is: metastatic, or relapsed/refractory to treatment, or locally advanced and not amenable to curative treatment, or limited disease treated with curative intent treatment within the last 2 years. This excludes non-melanoma skin cancers. * History of allergic reactions attributed to compounds of similar chemical or biologic composition to study agents. * Uncontrolled intercurrent illness (e.g., severe endocrinopathy, disseminated intravascular coagulation, profound electrolyte disturbance, active infectious hepatitis, uncontrolled dental infection) that in the opinion of the Site PI would make it unsafe to proceed with transplantation. INCLUSION CRITERIA: Donor * Related (age \>=12) and unrelated (age \>=18) donors deemed eligible (i.e., evaluated at NIH, COH, and FHCC in accordance with existing institutional Standard Policies and Procedures or evaluated per the standards required by the IRB of the National Marrow Donor Program or applicable registry), and willing to donate research samples will be included. * Ability of participant or parent/legal guardian to understand and the willingness to sign a written informed consent document. EXCLUSION CRITERIA: Donor None