Multi-Center Phase II Randomized Controlled Trial of Naïve T Cell Depletion for Prevention of Chronic Graft-Versus-Host Disease in Children and Young Adults

Inclusion Criteria: * The patient must have one of the following diagnoses and be considered to be an appropriate candidate for allogeneic HCT by the study site principal investigator (PI): * Acute lymphoblastic leukemia (ALL) with \< 5% marrow blasts. * Acute myeloid leukemia (AML) with \< 25% marrow blasts. * Other acute leukemia (OAL) or related neoplasm (including but not limited to acute biphenotypic leukemia \[ABL\], ambiguous lineage \[ALAL\], mixed phenotype acute leukemia \[MPAL\], blastic plasmacytoid dendritic cell neoplasm \[BPDCN\], acute undifferentiated leukemia \[AUL\], lymphoblastic lymphoma, Burkitt leukemia/lymphoma, mast cell leukemia, chronic monocytic leukemia \[CML\] with blast crisis or other chronic myeloproliferative neoplasm) with \< 5% marrow blasts. * Myelodysplastic syndrome (MDS) with excess blasts (EB-1 and EB-2) and has received cytotoxic induction chemotherapy (excluding small molecule inhibitors and de-methylating agents) * Age 6 months to 26 years at the time informed consent is obtained using the Informed Consent to Participate in a Research Study form * Matched related donor (MRD) or matched unrelated donor (MUD) (defined as 8/8 match for human leukocyte antigen \[HLA\]-A, -B, -C, -DRB1). * Planned product type for infusion is PBSC or BM (i.e. not cord blood): * For feasibility phase, planned product type for infusion must be PBSC. * For RCT, planned product type must be PBSC or BM. * Karnofsky or Lansky score \>= 60%. * Left ventricular ejection fraction (LVEF) at rest \>= 40%. * Diffusing capacity of the lungs for carbon monoxide (DLCO) (corrected for hemoglobin) \>= 60% predicted by pulmonary function tests (PFTs) \* Patients who are unable to perform PFTs (age \< 6 years or considered developmentally incapable of PFTs): oxygen saturation (by oximetry) must be \>= 92% on room air. * Total bilirubin =\< 2 x upper limit of normal (ULN) (unless value\[s\] \> 2 x ULN are disease- or medication-related). \* If value(s) are \> 2 x ULN and not disease- or medication related, patient must be evaluated by a gastrointestinal (GI) physician. If GI physician considers protocol treatment to be contraindicated for the patient, the patient will not be eligible for the study. * Alanine aminotransferase (ALT), aspartate aminotransferase (AST) =\< 2 x ULN (unless value\[s\] \> 2 x ULN are disease- or medication-related). \* If value(s) are \> 2 x ULN and not disease- or medication related, patient must be evaluated by a gastrointestinal GI physician. If GI physician considers protocol treatment to be contraindicated for the patient, the patient will not be eligible for the study. * Serum creatinine (SCr) within normal range for age. If SCr is outside normal range for age, creatinine clearance (CrCl) \> 40 mL/min/1.73m\^2 must be obtained (measured by 24-hour \[hr\] urine specimen or nuclear glomerular filtration rate \[GFR\]). * Age (Years): Maximum SCr (mg/dL) * =\< 5: 0.8 * 6-10: 1 * 11-15: 1.2 * \> 15: 1.5 * Recipient informed consent/assent/legal guardian permission documentation must be obtained. * DONOR: May be related (MRD) or unrelated (MUD) to the subject. * DONOR: Must be matched to the subject at 8/8 HLA alleles (HLA-A, -B, -C, and -DRB1) * DONOR: Be \>=14 years of age. * DONOR: Must be available to donate in the United States of America (USA) (i.e. excludes international donors). * DONOR: Must agree to donate BM or PBSC (i.e. agree to donate whichever product type is requested) (applicable only to the RCT phase of this study). * DONOR: MUDs: * Must give informed consent according to 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)) * Tests must be performed using Food and Drug Administration (FDA) licensed, cleared, and approved test kits in a Clinical Laboratory Improvement Amendments (CLIA)-certified laboratory * DONOR: MRDs: * Must be negative for human immunodeficiency virus (HIV)-1, HIV-2, human T-lymphotropic virus (HTLV)-1, HTLV-2, hepatitis B, hepatitis C (serological and/or nucleic acid testing \[NAT\] and/or other approved testing) * Must meet institutional donor eligibility criteria, or be ineligible with statement that the donor is a first or second degree relative (exception 21 CRF 1271.65(b)(i)). * Tests must be performed using FDA licensed, cleared, and approved test kits in a CLIA-certified laboratory. Exclusion Criteria: * Active central nervous system (CNS) disease. A patient may have a history of CNS disease; however, any CNS disease must be cleared by the end of the pre-conditioning evaluation. If CNS disease is identified on the first 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 the prevention of GVHD. * Patient body weight: * Matched related donor (MRD): \> 100 kg are ineligible * Matched unrelated donor (MUD): \> 75 kg must be discussed with the protocol principal investigator (PI) prior to enrollment. * HIV-positive. * Uncontrolled infections must be evaluated by an infectious disease physician and considered suitable to undergo HCT by the study site PI, infectious disease physician and protocol PI. Upper respiratory tract infection (URI) does not constitute an uncontrolled infection in this context. * Life expectancy \< 3 months from disease other than acute leukemia or myelodysplastic syndrome (MDS). * Significant medical condition that would make recipient unsuitable for HCT. * Prior allogeneic or autologous HCT. * Females who are pregnant or breastfeeding. * Patients of child bearing age who are presumed to be fertile and are unwilling to use an effective birth control method or refrain from sexual intercourse during study treatment and for 12 months following HCT. * Known hypersensitivity to tacrolimus, fludarabine, or methotrexate (MTX).