• 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

• 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.