-INCLUSION CRITERIA - Recipient

          1. 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)

          2. 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

          3. 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.

          4. Karnofsky performance score >=60

          5. Ability of subject to understand and the willingness to sign a written informed
             consent document.

          6. 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.

          7. 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.

          8. 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:

          1. 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.

          2. 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

          3. Uncontrolled intercurrent illness that in the opinion of the site PI would make it
             unsafe to proceed with transplantation.

          4. 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.

          5. 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