-INCLUSION CRITERIA - Recipient  Subjects must have a histologically or cytologically confirmed hematologic malignancy with standard indication for allogeneic hematopoietic cell transplantation including, but not limited to, one of the following:  Acute myeloid leukemia in morphologic complete remission (<5% blasts in the bone marrow, no detectable abnormal peripheral blasts, and no extramedullary disease) B-cell acute lymphoblastic leukemia in first or subsequent complete remission T-cell 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, 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 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 higher60 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 60-85 years, or age 18-60 years and unfit for myeloablative conditioning. Reasons for unfitness for myeloablative conditioning include:  Prior myeloablative HCT Prior exposure to inotuzumab, gemtuzumab, or other agent that increases the risk for sinusoidal obstruction syndrome. Significant organ dysfunction (e.g., creatinine or liver enzymes above the upper limit of normal or EGFR <=70 ml/min/1.73sq.m; prior sinusoidal obstruction syndrome, hepatic fibrosis, hepatic steatosis, or nodular regenerative hyperplasia; reduced ejection fraction <55% or focal hypokinesis, FEV1 or adjusted DLCO <75% of predicted) Hematopoietic Cell Transplantation- Comorbidity Index (HCT-CI) >= 3 Subject refusal of MAC (including subjects insistent on trying to maintain fertility) Pre-frail or frail by Fried s frailty phenotype Karnofsky performance score <80 Significant life-threatening toxicities associated with prior chemotherapy Co-morbidity considered by the treating physician to be exclusionary of MAC At least one potentially suitable HLA-matched related, HLA-haploidentical first degree or collateral related, HLA-matched unrelated, or >=5/10 HLA-mismatched unrelated donor. Karnofsky performance score >=60 Ability of subject to understand and the willingness to sign a written informed consent document.  Adequate organ function defined as possessing all of the following:  Cardiac ejection fraction >=35%; Forced expiratory volume-1, forced vital capacity, and diffusing capacity of the lung for carbon monoxide (corrected for hemoglobin) all of >=40% predicted; Serum creatinine clearance of >=45 ml/minute calculated using the Cockcroft-Gault equation; Total bilirubin <=2X the upper limit of normal; Alanine aminotransferase and aspartate aminotransferase <=5X the upper limit of normal.  Nonmyeloablative conditioning is toxic to the developing human fetus and is teratogenic. For this reason, the following measures apply:  Women of child-bearing potential (WOCBP) and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry and for at least one year post-transplant. Should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she should inform her treating physician immediately. WOCBP must have a negative serum or urine pregnancy test within 7 days prior to enrollment. For NIH treated subjects only: subjects requiring standard therapies to prepare for HCT should be referred in remission, if possible. However, these diseases are often aggressive and require swift evaluation for HCT while concurrently attempting to establish disease control through the administration of standard therapies. If ongoing therapy for the underlying disease outside of the NIH is not in the best interest of the subject according to the clinical judgment of the NIH PI, then the subject may receive standard treatment for his/her underlying hematologic malignancy as a bridge to HCT on this protocol, prior to starting the research phase of the study. If it becomes apparent that the subject will not be able to proceed to HCT, then he/she must come off study. Subjects receiving standard therapy will be told about the therapy, associated risks, potential benefits, alternatives to the proposed therapy, and the availability of receiving the same treatment elsewhere, outside of a research protocol.  EXCLUSION CRITERIA - Recipient:  Subjects 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.  Poorly controlled malignant indication for transplantation such as:  Leukemia not having achieved morphologic remission (i.e. bone marrow blasts >5% or active extramedullary disease) Lymphoma not having achieved at least a partial response to prior chemotherapy or radiation Uncontrolled intercurrent illness that in the opinion of the site PI would make it unsafe to proceed with transplantation. The potential for some of the study medications to be transmissible via breast milk of nursing mothers is unknown. Because there is unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother, breastfeeding must be discontinued. Active malignancy of non-hematopoietic type which is: metastatic, 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 nonmelanoma skin cancers.  INCLUSION CRITERIA - Related Donor:  Related donor (age >=12) deemed suitable and eligible, and willing to donate, per clinical evaluations, who are additionally willing to donate blood, bone marrow, and stool for research.  Related donors will be evaluated in accordance with existing institutional Standard Policies and Procedures for determination of eligibility and suitability for clinical donation.  Women of child-bearing potential (WOCBP) and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry and for at least 60 days after donation.  EXCLUSION CRITERIA - Related Donor:  None