Stratum 1 Recipient Inclusion Criteria:

1. Age ≥ 18 years and \< 66 years (chemotherapy-based conditioning) or \< 61 years (total body irradiation \[TBI\]-based conditioning) at the time of signing informed consent
2. Patient or legally authorized representative has the ability to provide informed consent according to the applicable regulatory and institutional requirements.
3. Stated willingness to comply with all study procedures and availability for the duration of the study.
4. Planned MAC regimen as defined per study protocol
5. Available partially HLA-MMUD (4/8-7/8 at HLA-A, -B, -C, and -DRB1 is required) with age ≥ 18 and ≤ 40 years (≤ 35 preferred).
6. Product planned for infusion is MMUD T-cell replete PBSC allograft
7. HCT-CI \< 5. The presence of prior malignancy will not be used to calculate HCT-CI for this trial to allow for the inclusion of patients with secondary or therapy-related AML or MDS.
8. One of the following diagnoses:

1. Acute myeloid leukemia (AML), acute lymphoblastic leukemia (ALL), or other acute leukemia in 1st remission or beyond with ≤ 5% marrow blasts and no circulating blasts or evidence of extra-medullary disease. Documentation of bone marrow assessment will be accepted within 45 days prior to the anticipated start of conditioning.
2. Patients with MDS with no circulating blasts and with \< 10% blasts in the bone marrow (higher blast percentage allowed in MDS due to lack of differences in outcomes with \< 5% or 5-10% blasts in MDS). Documentation of bone marrow assessment will be accepted within 45 days prior to the anticipated start of conditioning.
9. Cardiac function: Left ventricular ejection fraction ≥ 45% based on most recent echocardiogram or multi-gated acquisition scan (MUGA) results.
10. Estimated creatinine clearance ≥ 45mL/min calculated by equation.
11. Pulmonary function: diffusing capacity of the lungs for carbon monoxide (DLCO) corrected for hemoglobin \> 50% and forced expiratory volume in first second (FEV1) predicted \> 50% based on most recent pulmonary function test (PFT) results
12. Liver function acceptable per local institutional guidelines
13. KPS of ≥ 70%

Stratum 2 Recipient Inclusion Criteria:

1. Age ≥18 years at the time of signing informed consent
2. Patient or legally authorized representative has the ability to provide informed consent according to the applicable regulatory and local institutional requirements.
3. Stated willingness to comply with all study procedures and availability for the duration of the study.
4. Planned NMA/RIC regimen per study protocol
5. Available partially HLA-MMUD (4/8-7/8 at HLA-A, -B, -C, and -DRB1 is required) with age ≥ 18 and ≤ 40 years (≤ 35 preferred).
6. Product planned for infusion is MMUD T-cell replete PBSC allograft
7. One of the following diagnoses:

1. Patients with acute leukemia or chronic myeloid leukemia (CML) with no circulating blasts, no evidence of extramedullary disease, and with \< 5% blasts in the bone marrow.

Documentation of bone marrow assessment will be accepted within 45 days prior to the anticipated start of conditioning.
2. Patients with MDS with no circulating blasts and with \< 10% blasts in the bone marrow (higher blast percentage allowed in MDS due to lack of differences in outcomes with \< 5% or 5-10% blasts in MDS.) Documentation of bone marrow assessment will be accepted within 45 days prior to the anticipated start of conditioning.
3. Patients with chronic lymphocytic leukemia (CLL) or other leukemias (including prolymphocytic leukemia) with chemosensitive disease at time of transplantation
4. Higher risk CMML according to CMML-specific prognostic scoring system or high risk MDS/MPN not otherwise specified are eligible, provided there is no evidence of high-grade bone marrow fibrosis or massive splenomegaly at the time of enrollment.
5. Patients with lymphoma with chemosensitive disease at the time of transplantation
8. Cardiac function: Left ventricular ejection fraction ≥ 40% based on most recent echocardiogram or MUGA results with no clinical evidence of heart failure
9. Estimated creatinine clearance ≥ 45mL/min calculated by equation
10. Pulmonary function: DLCO corrected for hemoglobin \> 50% and FEV1 predicted \>50% based on most recent PFT results
11. Liver function acceptable per local institutional guidelines
12. KPS of ≥ 60%

Stratum 3 Recipient Inclusion Criteria:

1. Age ≥18 years at the time of signing informed consent
2. Patient or legally authorized representative has the ability to provide informed consent according to the applicable regulatory and local institutional requirements.
3. Stated willingness to comply with all study procedures and availability for the duration of the study.
4. Planned NMA/RIC regimen per study protocol
5. Available partially HLA-MMUD (4/8-7/8 at HLA-A, -B, -C, and -DRB1 is required) with age ≥ 18 and ≤ 40 years (≤ 35 preferred).
6. Product planned for infusion is MMUD T-cell replete PBSC allograft
7. Diagnosis of primary myelofibrosis with risk features making them eligible for HCT. Myelofibrosis secondary to essential thrombocythemia, polycythemia vera, or MDS with grade 4 fibrosis are also eligible. Patients with a myelofibrosis diagnosis require sponsor approval before enrolling.
8. Cardiac function: Left ventricular ejection fraction ≥ 40% based on most recent echocardiogram or MUGA results with no clinical evidence of heart failure
9. Estimated creatinine clearance ≥ 45 mL/min calculated by equation
10. Pulmonary function: DLCO corrected for hemoglobin \> 50% and FEV1 predicted \>50% based on most recent PFT results
11. Liver function acceptable per local institutional guidelines
12. KPS of ≥ 60%

Donor Inclusion Criteria (note: donors are not research subjects):

1. Must be unrelated to the subject and high-resolution HLA-matched at 4/8, 5/8, 6/8, or 7/8 (HLA-A, -B, -C, and -DRB1.
2. Donor must be typed at high-resolution for a minimum of HLA-A, -B, -C, -DQB1, and -DPB1.
3. Age ≥ 18 years and ≤ 40 years at the time of signing informed consent for PBSC donation. Note: donors are preferred to be ≤ 35.
4. Meet the donor registries' medical suitability requirements for PBSC donation.
5. Must undergo eligibility screening according to current Food and Drug Administration (FDA) requirements. Donors who do not meet one or more of the donor screening requirements may donate under urgent medical need.
6. Must agree to donate PBSC.
7. Must have the ability to give informed consent according to standard (non-study) informed consent according to applicable donor regulatory requirements.

Recipient Exclusion Criteria (Strata 1, 2, and 3):

1. Suitable HLA-matched related or 8/8 high-resolution matched unrelated donor available
2. Subject unwilling or unable to give informed consent, or unable to comply with the protocol including required follow-up and testing
3. Subjects with a prior allogeneic transplant
4. Subjects with an autologous transplant within the past 3 months
5. Females who are breast-feeding or pregnant
6. Uncontrolled bacterial, viral, or fungal infection at the time of the transplant preparative regimen
7. Concurrent enrollment on a preventative GvHD and/or infectious disease prevention clinical trial.
8. Subjects who undergo desensitization to reduce anti-donor HLA antibody levels prior to transplant.
9. Patients who are HIV+ with persistently positive viral load. HIV-infected patients on effective anti-retroviral therapy (ART) with undetectable viral load within 6 months are eligible for this trial. Patients with well controlled HIV are eligible provided resistance panels are negative, the patient is compliant with ART, and their disease remains well controlled.

Donor Exclusion Criteria:

1. Donor unwilling or unable to donate.
2. Recipient positive for HLA antibodies against a mismatched HLA in the selected donor determined by the presence of donor specific HLA antibodies (DSA) to any mismatched HLA allele/antigen at any of the following loci (HLA-A, -B, -C, -DRB1, DRB3, DRB4, DRB5, -DQA1, - DQB1, -DPA1, -DPB1) with median fluorescence intensity (MFI) \>3000 by microarray-based single antigen bead testing. In patients receiving red blood cell or platelet transfusions, DSA evaluation must be performed or repeated post-transfusion and prior to donor mobilization and initiation of recipient preparative regimen.