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. Planned MAC regimen as defined per protocol 3. Available partially HLA-MMUD (4/8-7/8 at HLA-A, -B, -C, and -DRB1 is required) with age \< 35 years 4. Product planned for infusion is PBSC 5. HCT Comorbidity Index (HCT-CI) \< 5 6. 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 myelodysplastic syndrome (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. 7. Cardiac function: Left ventricular ejection fraction \> 45% based on most recent echocardiogram or multigated acquisition scan (MUGA) results 8. Estimated creatinine clearance \> 60 mL/min calculated by equation 9. 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 results 10. Liver function acceptable per local institutional guidelines 11. Karnofsky performance status (KPS) of \> 70% 12. Subjects ≥ 18 years of age or legally authorized representative must have the ability to give informed consent according to applicable regulatory and local institutional requirements. Stratum 2 Recipient Inclusion Criteria 1. Age \> 18 years at the time of signing informed consent 2. Planned NMA/RIC regimen as defined per protocol 3. Available partially HLA-MMUD (4/8-7/8 at HLA-A, -B, -C, and -DRB1 is required) with age \< 35 years 4. Product planned for infusion is PBSC 5. 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. Patients with lymphoma with chemosensitive disease at the time of transplantation 6. Cardiac function: Left ventricular ejection fraction \> 45% based on most recent echocardiogram or MUGA results with no clinical evidence of heart failure 7. Estimated creatinine clearance \> 60 mL/min calculated by equation 8. Pulmonary function: DLCO corrected for hemoglobin \> 50% and FEV1 predicted \> 50% based on most recent pulmonary function test results 9. Liver function acceptable per local institutional guidelines 10. KPS of \> 60% 11. Subjects ≥ 18 years of age or legally authorized representative must have the ability to give informed consent according to applicable regulatory and local institutional requirements. Stratum 3 Recipient Inclusion Criteria 1. Age \> 1 years and \< 21 years at the time of signing informed consent 2. Partially HLA-MMUD (4/8-7/8 at HLA-A, -B, -C, and -DRB1 is required) with age \< 35 years 3. Product planned for infusion is BM 4. Planned MAC regimen as defined per protocol 5. One of the following diagnosis: 1. AML in 1st remission or beyond with ≤ 5% marrow blasts, no circulating blasts or evidence of extra-medullary disease. Pre-transplant MRD testing will be performed as per standard of practice at the treating institution. Patients with any MRD status are eligible and should be enrolled at the discretion of provider. Documentation of bone marrow assessment will be accepted within 45 days prior to the anticipated start of conditioning. 2. Patients MDS with no circulating blasts and less than 10% blasts in the bone marrow. Documentation of bone marrow assessment will be accepted within 45 days prior to the anticipated start of conditioning. 3. ALL in 1st remission or beyond with ≤ 5% marrow blasts, no circulating blasts, or evidence of extra-medullary disease. Pre-transplant MRD testing will be performed as standard practice at the treating institution with the goal of achieving MRD of \<0.01%. Patients with any MRD status are eligible and should be enrolled at the discretion of provider. Documentation of bone marrow assessment will be accepted within 45 days prior to the anticipated start of conditioning. 4. Other leukemia (mixed-phenotype acute leukemia \[MPAL\], CML, or other leukemia) in morphologic remission with ≤ 5% marrow blasts and no circulating blasts or evidence of extramedullary disease. Documentation of bone marrow assessment will be accepted within 45 days prior to the anticipated start of conditioning. 5. Chemotherapy sensitive lymphoma in at least partial remission (PR) 6. KPS or Lansky performance score ≥ 70% 7. Cardiac function: Left ventricular ejection fraction of ≥ 50% and shortening fraction of ≥ 27% based on most recent echocardiogram 8. Glomerular Filtration Rate (GFR) of ≥ 60ml/min/1.73m2 measured by nuclear medicine scan or calculated from a 24 hour urine collection 9. Pulmonary function: DLCO corrected for hemoglobin, FEV1, and Forced Vital Capacity (FVC) of ≥50% if able to perform pulmonary function tests. If unable to perform pulmonary function tests, must have a resting pulse oximetry of \>92% without supplemental oxygen. 10. Hepatic: Total bilirubin ≤ 2.5 mg/dL and alanine aminotransferase (ALT), aspartate aminotransferase (AST) \< 3x the upper limit of normal 11. Legal guardian permission must be obtained for subjects \< 18 years of age. Pediatric subjects will be included in age appropriate discussion in order to obtain assent. 12. Subjects ≥ 18 years of age or legally authorized representative must have the ability to give informed consent according to applicable regulatory and local institutional requirements. Donor Inclusion Criteria: 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, -DRB1, -DQB1, and -DPB1 3. Age \> 18 years and \< 35 years at the time of signing informed consent 4. Meet the donor registries' medical suitability requirements for PBSC or BM 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 (or BM for stratum 3) 7. Must have the ability to give 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. Primary myelofibrosis or myelofibrosis secondary to essential thrombocythemia, polycythemia vera, or MDS with grade 4 marrow fibrosis 4. Subjects with a prior allogeneic HSC transplant 5. Subjects with an autologous HSC transplant within the past 3 months 6. Females who are breast-feeding or pregnant 7. Uncontrolled bacterial, viral or fungal infection at the time of the transplant preparative regimen 8. Concurrent enrollment on other interventional GVHD clinical trial (enrollment on supportive care trials may be allowed after discussion with Principal Investigators) 9. Subjects who undergo desensitization to reduce anti-donor HLA antibody levels prior to transplant. 10. Patients who are HIV+ with persistently positive viral load. HIV-infected patients on effective anti-retroviral therapy with undetectable viral load within 6 months are eligible for this trial. Donor