Azacitidine, Venetoclax, and Gilteritinib in Treating Patients With Recurrent/Refractory FLT3-Mutated Acute Myeloid Leukemia, Chronic Myelomonocytic Leukemia, or High-Risk Myelodysplastic Syndrome/Myeloproliferative Neoplasm

Inclusion Criteria: * Diagnosis: * Phase I cohort: Adults ≥ 18 years with relapsed/refractory FLT3-mutated AML or CMML or other MDS/MPN that is intermediate-2 or high-risk by the International Prognostic Scoring System * Phase II cohort A: Adults ≥ 18 years with newly diagnosed FLT3-mutated AML * Phase II cohort B: Adults ≥ 18 years with relapsed/refractory FLT3-mutated AML or CMML or other MDS/MPN that is intermediate-2 or high-risk by the International Prognostic Scoring System * For all cohorts, patients with either FLT3-internal tandem duplication (FLT3-ITD) or FLT3 D835 mutations will be eligible * Performance status ≤ 3 (Eastern Cooperative Oncology Group \[ECOG\] scale) * Total serum bilirubin ≤ 2.5 x upper limit of normal (ULN), unless due to Gilbert's syndrome, hemolysis or the underlying leukemia approved by the principal investigator (PI) * Alanine aminotransferase (ALT) or aspartate aminotransferase (AST) ≤ 3 x ULN, unless due to the underlying leukemia approved by the PI * Creatinine clearance ≥ 30 mL/min * Ability to swallow * Signed informed consent Exclusion Criteria: * Prior therapies: * Phase I cohort: No restriction based on prior therapies * Phase II cohort A: Patients with prior therapy for AML are not eligible. Prior therapy for antecedent hematologic disorder is allowed. Prior hydroxyurea or cytarabine given for purposes of cytoreduction is also allowed. Prior all trans-retinoic acid given for presumed acute promyelocytic leukemia is also allowed * Phase II cohort B: No restriction on number of prior therapies * Patients suitable for and willing to receive intensive induction chemotherapy (for Phase II cohort A only) * Congenital long QT syndrome or corrected QT interval by Fridericia (QTcF) \> 450 msec. Repeat electrocardiograms (EKGs) after correction of electrolytes or discontinuation of QT prolonging medications are allowed to meet entry criteria. In cases where QTcF \> 450 msec is considered to be falsely increased due to inaccurate automated reading and not clinically significant (e.g. due to bundle branch block), patients are still eligible if cardiologist reviews and documents that QTcF is ≤ 450 msec when manually measured * Active serious infection not controlled by oral or intravenous antibiotics (e.g. persistent fever or lack of improvement despite antimicrobial treatment) * Active grade III-V cardiac failure as defined by the New York Heart Association criteria * Active central nervous system leukemia * Known human immunodeficiency virus (HIV) seropositive * Known hepatitis B surface antigen seropositive or known or suspected active hepatitis C infection * Note: Patients who have isolated positive hepatitis B core antibody (i.e., in the setting of negative hepatitis B surface antigen and negative hepatitis B surface antibody) must have an undetectable hepatitis B viral load. Patients who have positive hepatitis C antibody may be included if they have an undetectable hepatitis C viral load * Patients with a prior or concurrent malignancy whose natural history or treatment is not anticipated to interfere with the safety or efficacy assessment of the investigational regimen may be included only after discussion with the PI * Consumed strong inducer of cytochrome P450, family 3, subfamily A (CYP3A) or p-glycoprotein within 3 days of study enrollment. Agents include but are not limited to: carbamazepine, phenytoin, rifampin, and St. John's wort * Treatment with any investigational antileukemic agents or chemotherapy agents in the last 7 days before study entry, unless full recovery from side effects has occurred or patient has rapidly progressive disease judged to be life-threatening by the investigator. Prior recent treatment with corticosteroids, hydroxyurea and/or cytarabine (given for cytoreduction) permitted * Pregnant women will not be eligible; women of childbearing potential should have a negative pregnancy test prior to entering on the study and be willing to practice methods of contraception throughout the study period and for at least 6 months after the last dose of study drugs. Women do not have childbearing potential if they have had a hysterectomy or are postmenopausal without menses for 12 months. In addition, men enrolled on this study should understand the risks to any sexual partner of childbearing potential and should practice an effective method of birth control throughout the study period and for at least 4 months after the last dose of study drugs. Lactating women (or those planning to breastfeed) should not breastfeed during treatment of gilteritinib and for at least 2 months after the last dose of gilteritinib