Mismatched Related Donor Versus Matched Unrelated Donor Stem Cell Transplantation for Children, Adolescents, and Young Adults With Acute Leukemia or Myelodysplastic Syndrome

Inclusion Criteria: * PATIENT INCLUSION CRITERIA FOR ENROLLMENT: * 6 months to \< 22 years at enrollment * Diagnosed with ALL, AML, or MDS or mixed phenotype acute leukemia (MPAL) for which an allogeneic hematopoietic stem cell transplant is indicated. Complete Remission (CR) status will not be confirmed at the time of enrollment. CR as defined in these sections is required to proceed with the actual HCT treatment plan * Has not received a prior allogeneic hematopoietic stem cell transplant * Does not have a suitable human leukocyte antigen (HLA)-matched sibling donor available for stem cell donation * Has an eligible haploidentical related family donor based on at least intermediate resolution HLA typing * Patients who also have an eligible 8/8 MUD adult donor based on confirmatory high resolution HLA typing are eligible for randomization to Arm A or Arm B. * Patients who do not have an eligible MUD donor are eligible for enrollment to Arm C * All patients and/or their parents or legal guardians must sign a written informed consent * All institutional, Food and Drug Administration (FDA), and National Cancer Institute (NCI) requirements for human studies must be met * Co-Enrollment on other trials * Patients will not be excluded from enrollment on this study if already enrolled on other protocols for treatment of high risk and/or relapsed ALL, AML, MPAL and MDS. This is including, but not limited to, COG AAML1831, COG AALL1821, the EndRAD Trial, as well as local institutional trials. We will collect information on all co-enrollments * Patients will not be excluded from enrollment on this study if receiving immunotherapy prior to transplant as a way to achieve remission and bridge to transplant. This includes chimeric antigen receptor (CAR) T cell therapy and other immunotherapies * PATIENT INCLUSION CRITERIA TO PROCEED TO HCT: * Karnofsky Index or Lansky Play-Performance Scale \>= 60 on pre-transplant evaluation. Karnofsky scores must be used for patients \>= 16 years of age and Lansky scores for patients =\< 16 years of age (within 4 weeks of starting therapy) * A serum creatinine based on age/gender as follows: 6 months to \< 1 year: 0.5 mg/dL (Male); 0.5 mg/dL (Female) 1. to \< 2 years: 0.6 mg/dL (Male); 0.6 mg/dL (Female) 2. to \< 6 years: 0.8 mg/dL (Male); 0.8 mg/dL (Female) 6 to \< 10 years: 1 mg/dL (Male); 1 mg/dL (Female) 10 to \< 13 years: 1.2 mg/dL (Male); 1.2 mg/dL (Female) 13 to \< 16 years: 1.5 mg/dL (Male); 1.4 mg/dL (Female) \>= 16 years: 1.7 mg/dL (Male); 1.4 mg/dL (Female) * OR * A 24 hour urine Creatinine clearance \>= 60 mL/min/1.73 m\^2 * OR * A glomerular filtration rate (GFR) \>= 60 mL/min/1.73 m\^2. GFR must be performed using direct measurement with a nuclear blood sampling method OR direct small molecule clearance method (iothalamate or other molecule per institutional standard) * Note: Estimated GFR (eGFR) from serum creatinine, cystatin C or other estimates are not acceptable for determining eligibility * Serum glutamic-oxaloacetic transaminase (SGOT) aspartate aminotransferase \[AST\] or serum glutamate pyruvate transaminase (SGPT) aminotransferase \[ALT\] \< 5 x upper limit of normal (ULN) for age * Total bilirubin \< 2.5 mg/dL, unless attributable to Gilbert's Syndrome * Shortening fraction of \>= 27% by echocardiogram or radionuclide scan (MUGA) * OR * Ejection fraction of \>= 50% by echocardiogram or radionuclide scan (MUGA), choice of test according to local standard of care * Forced expiratory volume in 1 second (FEV1), forced vital capacity (FVC), and corrected carbon monoxide diffusing capability (DLCO) must all be \>= 50% of predicted by pulmonary function tests (PFTs). * For children who are unable to perform for PFTs (e.g., due to age or developmental delay), the criteria are: no evidence of dyspnea at rest, oxygen (O2) saturation (Sat) \> 92% on room air by pulse oximetry, not on supplemental O2 at rest, and not on supplemental O2 at rest * MPAL in first complete remission (CR1) for whom transplant is indicated. Examples include those patients who are poorly responsive to ALL therapy (end of induction failure( IF-MPAL) to ALL induction (see IF-MPAL note below), end of induction MRD ≥ 5% or end-of-consolidation MRD \> 0.01%), as well as patients treated with AML therapy * IF-MPAL: additional criterion for Induction failure for MPAL ONLY as per ALL1732: * An increasing number of circulating leukemia cells on 3 or more consecutive CBCs obtained at daily or longer intervals following day 8 of Induction therapy and prior to day 29 with confirmation by flow cytometry OR development of new sites of extramedullary disease, or other laboratory or clinical evidence of refractory disease or progression prior to the end of Induction evaluation (note that residual testicular disease at the end of Induction is an exception) * MPAL in \> second complete remission (CR2) * ALL high-risk in CR1 for whom transplant is indicated. Examples include: induction failure, treatment failure as per minimal residual disease by flow cytometry \> 0.01% after consolidation and not eligible for AALL1721 or AALL1721 not available/unwilling to enroll, hypodiploidy (\< 44 chromosomes) with MRD+ \> 0.01% after induction, persistent or recurrent cytogenetic or molecular evidence of disease during therapy requiring additional therapy after induction to achieve remission (e.g. persistent molecular BCR-ABL positivity), T cell ALL with persistent MRD \> 0.01% after consolidation. * ALL in CR2 for whom transplant is indicated. Examples include: B-cell: early (=\< 36 months from initiation of therapy) bone marrow (BM) relapse, late BM relapse (\>= 36 months) with MRD \>= 0.1% by flow cytometry after first re-induction therapy; T or B-cell: early (\< 18 months) isolated extramedullary (IEM), late (\>= 18 months) IEM, end-Block 1 MRD \>= 0.1%; T-cell or Philadelphia chromosome positive (Ph+): BM relapse at any time * ALL in \>= third complete remission (CR3) * Patients treated with chimeric antigen receptor T-cells (CART) cells for whom transplant is indicated. Examples include: transplant for consolidation of CART, loss of CART persistence and/or B cell aplasia \< 6 months from infusion or have other evidence (e.g., MRD+) that transplant is indicated to prevent relapse * AML in CR1 for whom transplant is indicated. Examples include those deemed high risk for relapse as described in AAML1831: * FLT3/ITD+ with allelic ratio \> 0.1 without bZIP CEBPA, NPM1 * FLT3/ITD+ with allelic ratio \> 0.1 with concurrent bZIP CEBPA or NPM1 and with evidence of residual AML (MRD \>= 0.05%) at end of Induction * Presence of RAM phenotype or unfavorable prognostic markers (other than FLT3/ITD) per cytogenetics, fluorescence in situ hybridization (FISH), next generation sequencing (NGS) results, regardless of favorable genetic markers, MRD status or FLT3/ITD mutation status * AML without favorable or unfavorable cytogenetic or molecular features but with evidence of residual AML (MRD \>= 0.05%) at end of Induction * Presence of a non-ITD FLT3 activating mutation and positive MRD (\>= 0.05%) at end of Induction 1 regardless of presence of favorable genetic markers. * AML in \>= CR2 * MDS with \< 5% blasts by morphology and flow cytometry (if available) on the pre-transplant bone marrow evaluation * Complete remission (CR) is defined as \< 5% blasts by morphology and flow cytometry (if available) on the pre-transplant bone marrow evaluation with minimum sustained absolute neutrophil count (ANC) of 300 cells/microliter for 1 week or ANC \> 500 cells/microliter. We will be collecting data from all approaches to MRD evaluation performed including NGS and polymerase chain reaction (PCR). It is strongly recommended that MPAL be evaluated using multidimensional flow cytometry and/or (KMT2Ar) qt PCR. It is strongly recommended that MPAL be evaluated using multidimensional flow cytometry and/or (KMT2Ar) qt PCR * DONOR ELIGIBILITY CRITERIA: * Matched Unrelated Donors: Unrelated donor candidates must be matched at high resolution at a minimum of 8/8 alleles (HLA-A, -B, -C, -DRB1). One-antigen HLA mismatches are not permitted. HLA matching of additional alleles is recommended according to National Marrow Donor Program (NMDP) guidelines, but will be at the discretion of local centers * Haploidentical Matched Family Members: * Minimum match level full haploidentical (at least 5/10; HLA-A, -B, -C, -DRB1, -DQB1 alleles). The following issues (in no particular order) should be considered in choosing a haploidentical donor: * Absent or low patient donor-specific antibodies (DSA) * Mean fluorescence intensity (MFI) of any anti-donor HLA antibody by solid phase immunoassay should be \< 2000. Donors with higher levels are not eligible. * If a screening assay against pooled HLA antigens is used, positive results must be followed with specificity testing using a single antigen assay. The MFI must be \< 2000 unless the laboratory has validated higher threshold values for reactivity for HLA antigens (such as HLA-C, -DQ, and -DP), that may be enhanced in concentration on the single antigen assays. Donor anti- recipient antibodies are of unknown clinical significance and do not need to be sent or reported. * Consult with Study Chair for the clinical significance of any recipient anti-donor HLA antibody. * If centers are unable to perform this type of testing, please contact the Study Chair to make arrangements for testing. * If killer immunoglobulin testing (KIR) is performed: KIR status by mismatch, KIR-B, or KIR content criteria can be used according to institutional guidelines. * ABO compatibility (in order of priority): * Compatible or minor ABO incompatibility * Major ABO incompatibility * CMV serostatus: * For a CMV seronegative recipient: the priority is to use a CMV seronegative donor when feasible * For a CMV seropositive recipient: the priority is to use a CMV seropositive donor when feasible * Age: younger donors including siblings/half-siblings, and second degree relatives (aunts, uncles, cousins) are recommended, even if \< 18 years * Size and vascular access appropriate by center standard for peripheral blood stem cell (PBSC) collection if needed * Haploidentical matched family members: screened by center health screens and found to be eligible * Unrelated donors: meet eligibility criteria as defined by the NMDP or other unrelated donor registries. If the donor does not meet the registry eligibility criteria but an acceptable eligibility waiver is completed and signed per registry guidelines, the donor will be considered eligible for this study * Human immunodeficiency virus (HIV) negative * Not pregnant * MUD donors and post-transplant cyclophosphamide haplo donors should be asked to provide BM. If donors refuse and other donors are not available, PBSC is allowed. TCR-alpha beta/CD19 depleted haplo donors must agree to donate PBSC * Must give informed consent: * Haploidentical matched family members: Institution standard of care donor consent and Protocol-specific Donor Consent for Optional Studies * Unrelated donors: standard NMDP Unrelated Donor Consent Exclusion Criteria: * PATIENT EXCLUSION CRITERIA FOR ENROLLMENT: * Patients with genetic disorders (generally marrow failure syndromes) prone to secondary AML/ALL/MPAL with known poor outcomes because of sensitivity to alkylator therapy and/or TBI are not eligible (Fanconi Anemia, Kostmann Syndrome, Dyskeratosis Congenita, etc). Patients with Downs syndrome because of increased toxicity with intensive conditioning regimens. * Patients with any obvious contraindication to myeloablative HCT at the time of enrollment * Female patients who are pregnant are ineligible as many of the medications used in this protocol could be harmful to unborn children and infants * Sexually active patients of reproductive potential who have not agreed to use an effective contraceptive method for the duration of their study participation * PATIENT EXCLUSION CRITERIA TO PROCEED TO HCT: * Patients with uncontrolled fungal, bacterial, viral, or parasitic infections are excluded. Patients with history of fungal disease during chemotherapy may proceed if they have a significant response to antifungal therapy with no or minimal evidence of disease remaining by computed tomography (CT) evaluation * Patients with active central nervous system (CNS) leukemia or any other active site of extramedullary disease at the time of initiation of the conditioning regimen are not permitted. * Note: Those with prior history of CNS or extramedullary disease, but with no active disease at the time of pre-transplant workup, are eligible * Pregnant or breastfeeding females are ineligible as many of the medications used in this protocol could be harmful to unborn children and infants