Combination Chemotherapy (FLAG-Ida) Followed Immediately by Reduced-Intensity Total Body Radiation Therapy and Donor Hematopoietic Cell Transplant for the Treatment of Adults Age 60 and Older With Newly Diagnosed Adverse-Risk Acute Myeloid Leukemia or Other High-Grade Myeloid Cancer

Inclusion Criteria: * PARTICIPANTS: Age ≥ 60 years. Adults age \< 60 years are eligible if they are felt to be unsuitable candidates for myeloablative conditioning as per physician assessment * PARTICIPANTS: Ability to understand and willingness to sign a written informed consent document. Legally authorized representatives may sign and give informed consent on behalf of potential study participants * PARTICIPANTS: Newly diagnosed, untreated high-risk myeloid or mixed myeloid/lymphoid neoplasm: * Adverse-risk AML (using 2022 International Consensus Classification for disease categorization and 2022 European LeukemiaNet \[ELN\] criteria for molecular/cytogenetic risk assignment) * Acute leukemia of ambiguous lineage (using 2022 International Consensus Classification for disease categorization) * High-risk myelodysplastic neoplasm (MDS) (Molecular International Prognostic System \[IPSS-M\] moderate high, high, or very high, OR ≥ 10% blasts in blood or marrow) * High-risk chronic myelomonocytic leukemia (CMML) (clinical/molecular CMML-specific prognostic scoring system \[CPSS-Mol\] intermediate-2 or high, OR ≥ 10% blasts in blood or marrow) * Prior treatment of MDS or CMML with lower-intensity therapy (e.g., growth factors, erythropoiesis-stimulating agents, and lenalidomide) is permissible, but patients may not have received prior hypomethylating agents * PARTICIPANTS: Disease not requiring immediate anti-neoplastic therapy (e.g., presenting with leukopenia or pancytopenia), defined as a clinical scenario in which delay of systemic leukemia-directed treatment would be unsafe. Supportive cytoreduction with hydroxyurea for transient disease control is allowed, and does not constitute immediate anti-neoplastic treatment * PARTICIPANTS: Interest in pursuing allogeneic HCT * PARTICIPANTS: Available caregiver * PARTICIPANTS: Karnofsky score ≥ 70; Eastern Cooperative Oncology Group (ECOG) performance status 0-2 * PARTICIPANTS: Bilirubin ≤ 2.5 x institutional upper limit of normal unless elevation is thought to be due to hepatic infiltration by myeloid neoplasm, Gilbert's syndrome, or hemolysis * PARTICIPANTS: Serum creatinine ≤ 1.5 mg/dL * PARTICIPANTS: Prior autologous HCT is permissible if \> 6 months after planned HCT on this study * PARTICIPANTS: Participants of child-bearing potential must be willing to employ two highly effective and acceptable forms of contraception 7 days before initiation of study treatment and for at least 12 months after HCT. Women of childbearing potential (WOCBP) must have a negative serum pregnancy test during screening (within 2 weeks of treatment initiation, per Fred Hutch Cancer Center \[FHCC\] standard of care \[SOC\]), where WOCBP are defined as all female participants between 18-55 years of age, unless postmenopausal or with hysterectomy * PARTICIPANTS: HIV-infected patients on effective anti-retroviral therapy with undetectable viral load within 6 months are eligible for this trial * PARTICIPANTS: For patients with evidence of chronic hepatitis B virus (HBV) infection, the HBV viral load must be undetectable on suppressive therapy, if indicated. Patients with a history of hepatitis C virus (HCV) infection must have been treated and cured. Patients with HCV infection who are currently on treatment are eligible if they have an undetectable HCV viral load * PARTICIPANTS: Patients with a prior or concurrent malignancy whose natural history or treatment does not have the potential to interfere with the safety or efficacy assessment of the investigational regimen, as determined by the investigator, are eligible for this trial * DONORS: Patients must have an HLA-matched related donor, an HLA-matched or mismatched unrelated donor, or an HLA- haploidentical donor who meets standard Fred Hutchinson Cancer Center (FHCC) and/or National Marrow Donor Program (NMDP) or other donor center criteria for peripheral blood stem cell (PBSC) donation as follows: * HLA-matched related donor: related to the patient and genotypically or phenotypically identical for HLA-A, B, C, DRB1 and DQB1. Phenotypic identity must be confirmed by high-resolution typing * HLA-matched unrelated donor: * 10/10 matched for HLA-A, B, C, DRB1 and DQB1 by high resolution typing * Donors are excluded when preexisting immunoreactivity is identified that would jeopardize donor hematopoietic cell engraftment. The recommended procedure for patients with 10 of 10 HLA allele level (phenotypic) match is to obtain panel reactive antibody (PRA) screens to class I and class II antigens for all patients before HCT. If the PRA shows \> 10% activity, then flow cytometric or B and T cell cytotoxic cross matches should be obtained. The donor should be excluded if any of the cytotoxic cross match assays are positive. A positive anti-donor cytotoxic crossmatch is an absolute donor exclusion * HLA-mismatched unrelated donor: * HLA-matching must be based on results of high resolution typing at HLA-A, -B, -C, -DRB1, and -DQ * Mismatched for a single allele without antigen mismatching at HLA-A, B, or C as defined by high resolution typing but otherwise matched for HLA-A, B, C, DRB1 and DQB1 by high resolution typing * HLA class I HLA-A, -B, -C allele matched donors allowing for any one or two DRB1 and/or DQB1 antigen/allele mismatch * Donor/recipient HLA mismatching at loci for which the patient is homozygous is not allowed (isolated rejection vector) * HLA-haploidentical donor: * Donors must be haploidentical relatives of the patients. Donor-recipient compatibility will be tested through HLA typing at high resolution for the HLA loci (-A, -B, -C, -DRB1, -DQB1). Donor and recipient should share at least 5/10 HLA loci * Donor age ≥ 12 years * Donor weight ≥ 40 kg * Ability of donors younger than 18 years of age to undergo apheresis without use of a vascular access device. Vein check must be performed and verified by an apheresis nurse prior to arrival * Donor must meet the selection criteria as defined by the Foundation of the Accreditation of Cell Therapy (FACT) and will be screened per the American Association of Blood Banks (AABB) guidelines * In case of more than one available haploidentical donor, preference should be given to younger age * Since detection of anti-donor-specific antibodies (anti-DSA) is associated with higher graft rejection rate, patients will be screened for anti-DSA pre-transplant. Use of donors requiring desensitization treatment of the patient are not permissible Exclusion Criteria: * PARTICIPANTS: Active central nervous system (CNS) disease * PARTICIPANTS: Decompensated congestive heart failure and/or uncontrolled arrhythmia and/or significant medical history of cardiac disease precluding allogeneic HCT * PARTICIPANTS: Significant medical history of pulmonary disease and/or symptoms suggestive of pulmonary disease precluding allogeneic HCT * PARTICIPANTS: Treatment with any other approved or investigational anti-leukemia agent(s) at the time of initiation of study treatment * PARTICIPANTS: Concomitant illness associated with a likely survival of \< 1 year * PARTICIPANTS: Active systemic fungal, bacterial, viral, or other infection, unless disease is under treatment with antimicrobials and/or controlled or stable. Patients with fever thought to be secondary to myeloid malignancy are eligible * PARTICIPANTS: Known hypersensitivity or contraindication to receiving any of the study drugs used in this trial, including post-transplant cyclophosphamide (PTCy) * PARTICIPANTS: Pregnancy or lactation * PARTICIPANTS: Psychiatric illness/social situations that would limit compliance with study requirements