Graft Versus Host Disease-Reduction Strategies for Donor Blood Stem Cell Transplant Patients With Acute Leukemia or Myelodysplastic Syndrome (MDS)

Inclusion Criteria: * Patients who are considered appropriate candidates for myeloablative, TBI-containing allogeneic hematopoietic stem cell transplantation and have one of the following diagnoses: * 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. * Patient age 1-60 years old (inclusive) at the time of informed consent * Patients aged 1-50 years old (inclusive) are eligible for TBI-based conditioning regimens. * Patients aged 1-60 years old (inclusive) are eligible for busulfan-based conditioning regimens (with or without TBI 4 Gy). * Patient with an HLA-matched (HLA-A, B, C, DRB1, and DQB1 matched) related or unrelated donor capable of donating PBSC. * Recipient informed consent/assent and/or legal guardian permission must be obtained. * DONOR: HLA-matched related and unrelated donors (HLA-A, B, C, DRB1 and DQB1 matched based on high-resolution typing). * DONOR: \>= 18 years old. * DONOR: Willing to donate PBSC. * DONOR: Matched related donors: * Must give informed consent using the related donor informed consent form. * Must meet institutional donor eligibility criteria or be ineligible with statement that the donor is a first or second degree relative (exception 21 Code of Federal Regulations \[CFR\] 1271.65(b)(i)). * DONOR: Matched unrelated donors: * Must consent according to the applicable National Marrow Donor Program (NMDP) donor regulatory requirements. * Must meet eligibility criteria as defined by the NMDP or be ineligible with statement of urgent medical need (exception 21 CFR 1271.65(b)(iii)). Exclusion Criteria: * Patients with central nervous system (CNS) involvement refractory to intrathecal chemotherapy and/or standard cranial-spinal radiation. A patient may have a history of CNS disease. However, any CNS disease must be cleared by the end of the pre-conditioning evaluation time frame. If CNS disease is identified on cerebrospinal fluid (CSF) evaluation within 30 days of the start of the preparative regimen a repeat CSF evaluation must be performed and show no evidence of disease in order for the patient to be eligible for the protocol. * Patients on other experimental protocols for prevention of GVHD. * Patient weight: * Patients with HLA-matched related donors will be excluded if they weigh \>= 110 kg. * Patients with HLA-matched unrelated donors will be excluded if they weigh \>= 110 kg and must be discussed with the Fred Hutch protocol principal investigator (PI) if they weigh \>= 90 kg. * Patients who are positive for human immunodeficiency virus (HIV)-1, HIV-2, human T-cell lymphotropic virus (HTLV)1 or HTLV2. * Patients with uncontrolled infections for whom myeloablative HCT is considered contraindicated by the consulting infectious disease physician; i.e. patients with active infections require infectious disease consultation and documentation by the infectious disease team that myeloablative HCT is not considered to be contraindicated. Upper respiratory tract infection is not considered to represent an uncontrolled infection in this context. * Patients with organ dysfunction, including: * Renal insufficiency (creatinine \> 1.5 mg/dl) at the time of evaluation for the protocol. Patients with a known history of creatinine \> 1.5 mg/dl or a current serum creatinine above the normal range for age must have a current creatinine clearance of \> 60 ml/min/1.73 m\^2 (measured by 24-hr urine specimen or nuclear glomerular filtration rate \[GFR\]). * Left ventricular ejection fraction \< 45%. * Carbon monoxide diffusing capability (DLCO) corrected \< 60%. Patients who are unable to perform pulmonary function tests (for example, due to young age and/or developmental status) will be excluded if the oxygen (O2) saturation is \< 92% on room air. * Liver function abnormality. Patients who have liver function test (LFT)s (specifically, total bilirubin, aspartate aminotransferase \[AST\] or alanine aminotransferase \[ALT\]) \>= twice the upper limit of normal should be evaluated by a gastrointestinal (GI) physician unless there is a clear precipitating factor (such as an azole, MTX, trimethoprim-sulfamethoxazole, or another drug). If the GI physician considers that HCT on the protocol is contraindicated, that patient will be excluded from the protocol. Patients with Gilbert's syndrome require GI physician consultation but may be included on the protocol. Patients with no other known liver function abnormality or with reversible drug-related transaminitis do not necessarily require GI consultation and may be included on the protocol. * Patients who have received previous myeloablative allogeneic or autologous transplantation. * Patients with a life expectancy \< 12 months from co-existing disease other than the leukemia or MDS. * Patients who are pregnant or breast-feeding. * Patients of childbearing age who are presumed to be fertile and are unwilling to use an effective birth control method or refrain from sexual intercourse during and for 12 months post-HCT. * Patients with any other significant medical conditions that would make them unsuitable for transplantation, as determined by the PI. * Patients with a known hypersensitivity to tacrolimus or MTX * Patients who have received checkpoint inhibitors within three months of transplantation unless an exception is made by the PI * DONOR: Donors who are HIV-1, HIV-2, HTLV-1, HTLV-2 seropositive or with active hepatitis B or hepatitis C virus infection. Test must be performed using Food and Drug Administration (FDA) licensed, cleared, and approved test kits (serological and/or nucleic acid amplification test \[NAT\] and/or other approved testing) in a Clinical Laboratory Improvement Act (CLIA)-certified laboratory. * Unrelated donors donating outside of the USA.