Depletion of CD45RA Positive Naive T Cells for the Prevention of Graft Versus Host Disease in Older Patients Undergoing Allogeneic Hematopoietic Cell Transplantation for Acute Leukemia or Myelodysplastic Syndrome

Inclusion Criteria: * PARTICIPANT: Age \> 60 years * PARTICIPANT: Diagnosis of acute leukemia or myelodysplastic syndrome (MDS): * Acute lymphocytic leukemia (ALL) in first or subsequent morphological remission (\< 5% marrow blasts by morphology). * Acute myeloid leukemia (AML) in first or subsequent morphological remission (\< 5% marrow blasts by morphology). * Other acute leukemia or related neoplasm (including but not limited to 'mixed phenotype' 'biphenotypic', 'acute undifferentiated' or 'ambiguous lineage' acute leukemia, blastic plasmacytoid dendritic cell neoplasm, lymphoblastic lymphoma, Burkitt leukemia/lymphoma, mast cell leukemia, chronic myeloid leukemia \[CML\] with blast crisis or other chronic myeloproliferative neoplasm) in first or subsequent morphological remission (\< 5% marrow blasts by morphology). * Myelodysplastic syndrome (MDS) with a history of excess blasts (≥ approximately 5% in marrow blasts by morphology) and a history of receiving cytoreductive therapy (including but not limited to BCL-2 inhibitors or cytotoxic chemotherapy) within the past 3 months * PARTICIPANT: Karnofsky performance status (KPS) ≥ 60% at pre-HCT evaluation * PARTICIPANT: Ability to provide informed consent * PARTICIPANT: Prior autologous or allogeneic HCT is permitted * DONOR: Human leukocyte antigen (HLA) matching: HLA-identical related donors or unrelated donors matched for HLA-A, -B, -C, - DRB1, and -DQB1 by high-resolution deoxyribonucleic acid (DNA) typing. A single allele-level mismatch at HLA class I is permitted * DONOR: Stem cell source: Donors must be able to undergo peripheral blood stem cell (PBSC) collection. Only G-CSF-mobilized PBSC are permitted as the hematopoietic stem cell (HSC) source on this protocol Exclusion Criteria: * PARTICIPANT: Circulating blasts: * Acute leukemia: circulating abnormal blasts detectable by standard pathology. * MDS: \> 5% circulating abnormal blasts by standard pathology * PARTICIPANT: Patients with promyelocytic leukemia (APL) * PARTICIPANT: Patients with active extramedullary disease are excluded. History of extramedullary disease is allowed if only minimal residual disease is present prior to HCT * PARTICIPANT: Ejection fraction \< 35% (or shortening fraction \< 26% if ejection fraction \[EF\] unavailable) * PARTICIPANT: Cardiac insufficiency requiring treatment or symptomatic coronary artery disease * PARTICIPANT: Patients with shortening fraction \< 26% may enroll if approved by a cardiologist * PARTICIPANT: Carbon monoxide diffusing capability (DLCO) \< 40%, total lung capacity (TLC) \< 40%, forced expiratory volume in 1 second (FEV1) \< 40%, and/or requiring continuous supplemental oxygen * PARTICIPANT: If pulmonary function tests (PFTs) cannot be obtained: any patient with room air oxygen saturation \< 89% during a 6-minute walk test (6MWT) will be excluded * PARTICIPANT: Serum creatinine must be within institutional normal limits * PARTICIPANT: If serum creatinine \> upper limit of normal, a 24-hour creatinine clearance will be performed; patients are excluded if \< 50 mL/min/1.73 m\^2 * PARTICIPANT: Exclude patients with any of the following: * Fulminant liver failure. * Cirrhosis with evidence of portal hypertension. * Alcoholic hepatitis. * Esophageal varices or history of variceal bleeding. * Hepatic encephalopathy. * Uncorrectable synthetic dysfunction (prolonged prothrombin time). * Ascites due to portal hypertension. * Bridging fibrosis. * Bacterial or fungal liver abscess. * Biliary obstruction. * Chronic viral hepatitis with total bilirubin \>3 mg/dL. * Symptomatic biliary disease * PARTICIPANT: Active infectious disease requiring deferral of conditioning (per Infectious disease consult). Upper respiratory tract infection is not considered to represent an uncontrolled infection in this context * PARTICIPANT: Fungal infection with radiologic progression despite appropriate antifungal therapy * PARTICIPANT: Viral infection risk: HIV-1, HIV-2, human T-lymphotropic virus (HTLV) 1 or HTLV2 positivity or active infectious hepatitis * PARTICIPANT: Patients with active central nervous system (CNS) leukemia at the time of treatment are excluded. A history of CNS leukemia is allowed if CNS disease has resolved prior to treatment initiation * PARTICIPANT: Weight \> 110kg * PARTICIPANT: Other malignancies: * Active non-hematologic malignancies (except non-melanoma skin cancers). * Non-hematologic malignancy in remission but with \> 20% risk of recurrence within 5 years. * Exclusion does not apply to indolent non-hematologic malignancies not requiring therapy * PARTICIPANT: Patients with a life expectancy \< 12 months from co-existing disease other than the leukemia or MDS * PARTICIPANT: Hypersensitivity to cyclophosphamide, fludarabine, tacrolimus or mycophenolate mofetil (MMF) * PARTICIPANT: Inability to provide informed consent * PARTICIPANT: Alternative treatment: Patients who are suitable for and willing to receive a standard high intensity (myeloablative) preparative regimen * DONOR: Stem cell source restriction: Donors (or centers) that can exclusively provide bone marrow harvests * DONOR: Medical contraindications: HIV-1, HIV-2, HTLV-1, HTLV-2 seropositive or with active hepatitis B or hepatitis C virus infection * DONOR: Preexisting immunoreactivity: * Determined per institutional standard practices. * For patients with a 10/10 HLA allele-level match, panel reactive antibody (PRA) screens to class I and class II antigens are recommended before HCT. If PRA \>10%, then flow cytometric or B- and T-cell cytotoxic crossmatches must be obtained. The donor is excluded if any crossmatch is positive. * For patients with a Class I allele mismatch, flow cytometric or B- and T-cell cytotoxic crossmatches must be performed regardless of PRA results. A positive anti-donor cytotoxic crossmatch is an absolute exclusion. * Vector homozygosity: Donors are excluded if the patient is homozygous in the graft rejection vector against the donor's mismatched HLA class I allele * DONOR: Donors donating outside of the United States of America (USA)