Multi-antigen Specific CD8+ T Cells With Decitabine and Lymphodepleting Chemotherapy for the Treatment of Patients With Relapsed or Refractory AML or MDS Following an Allogeneic Hematopoietic Cell Transplantation From a Matched Donor

Inclusion Criteria: * PARTICIPANT: Documented informed consent of the participant and/or legally authorized representative and documented informed consent of the donor * PARTICIPANT: Agreement to allow the use of archival tissue from diagnostic tumor biopsies (if unavailable, exceptions may be granted with study principal investigator \[PI\] approval) * PARTICIPANT: Age: ≥ 18 years * PARTICIPANT: Eastern Cooperative Oncology Group (ECOG) ≤ 1 or Karnofsky performance score (KPS) ≥ 70 * PARTICIPANT: Confirmed diagnosis of AML/MDS that has relapsed after or is refractory to an allogeneic hematopoietic cell transplantation (HCT) from a matched donor. * Refractory - failure to achieve a complete response minimal residual disease (CRMRD) (-) by multicolor flow cytometry (MFC) or reverse transcription polymerase chain reaction (RT qPCR) * Relapse - detection of clonal abnormal myeloid blasts by morphology (morphologic relapse) or by MFC, or RT-qPCR analysis (MRD\[+\] relapse) after achieving a CRMRD(-) induced by an allogeneic HCT or maintained by allogeneic HCT administered as consolidation therapy. Note: Patients who meet the protocol definition of relapse/refractory (r/r) AML/MDS at screening and subsequently achieve a CRMRD(-) response status following protocol-specified bridging therapy will remain eligible to continue participation in this study * PARTICIPANT: At least 100 days post allogeneic HCT * PARTICIPANT: Donor match at 8 out of 8 loci for human leucocyte antigen (HLA) -A, -B, -C, and -DRB1 (each typed at high resolution by deoxyribonucleic acid \[DNA\]-based methods) * PARTICIPANT: Expression of HLA-A\*0201 as determined by high resolution sequence-based typing methods * PARTICIPANT: Total bilirubin ≤ 1.5 X upper limit of normal (ULN) (unless has Gilbert's disease) (to be performed within 28 days of consenting) * PARTICIPANT: Aspartate aminotransferase (AST) ≤ 2.5 x ULN (to be performed within 28 days of consenting) * PARTICIPANT: Alanine aminotransferase (ALT) ≤ 2.5 x ULN (to be performed within 28 days of consenting) * PARTICIPANT: Serum creatinine \< 1.5 mg/dL or creatinine clearance of ≥ 60 mL/min per 24 hour urine test or the Cockcroft-Gault formula (to be performed within 28 days of consenting) * PARTICIPANT: Left ventricular ejection fraction (LVEF) ≥ 50% Note: To be performed before the first dose of lymphodepletion chemotherapy * PARTICIPANT: If able to perform pulmonary function tests: Forced expiratory volume in one second (FEV1), forced vital capacity (FVC), and diffuse lung capacity for carbon monoxide (DLCO) (diffusion capacity) ≥ 50% of predicted (corrected for hemoglobin) (to be performed within 28 days of consenting) * PARTICIPANT: If unable to perform pulmonary function tests: Oxygen (O2) saturation \> 92% on room air (to be performed within 28 days of consenting) * PARTICIPANT: Seronegative for HIV antigen/antibody (Ag/Ab) combo, hepatitis C virus (HCV), active hepatitis B virus (HBV) (surface antigen negative), and syphilis (rapid plasma reagin \[RPR\]) (to be performed within 28 days of consenting) * If positive, hepatitis C ribonucleic acid (RNA) quantitation must be performed. OR * If seropositive for HIV, HCV or HBV, nucleic acid quantitation must be performed. Viral load must be undetectable * PARTICIPANT: Women of childbearing potential (WOCBP): Negative urine or serum pregnancy test. If the urine test is positive or cannot be confirmed as negative, a serum pregnancy test will be required (to be performed within 28 days of consenting) * PARTICIPANT: Agreement by females and males of childbearing potential to use an effective method of birth control or abstain from heterosexual activity for the course of the study through at least 3 months after the completion of the last cycle of protocol therapy. * Childbearing potential defined as not being surgically sterilized (men and women) or have not been free from menses for \> 1 year (women only) * DONOR: The identified donor must be the donor whose stem cells were used for the research participant's allo HCT * DONOR: The donor's hematocrit value is ≥ 35% * DONOR: The donor's platelet count is \> 100,000 per microliter * CRITERIA TO PROCEED TO START OF CYCLE 1: NEXI-001 product is released from manufacturing with a certificate of analysis (COA) * CRITERIA TO PROCEED TO START OF CYCLE 1: Bone marrow aspirate and bone marrow biopsy within one week prior to treatment for baseline disease status (all disease statuses are eligible to proceed) and correlative studies * CRITERIA TO PROCEED TO START OF CYCLE 1: T-cell chimerism ≥ 50% to donor by polymerase chain reaction (PCR) analysis * CRITERIA TO PROCEED TO START OF CYCLE 1: Fully recovered to ≤ grade 1 from non-hematologic acute toxic effects (except alopecia) from prior anti-cancer therapy * CRITERIA TO PROCEED TO START OF CYCLE 1: Total bilirubin ≤ 1.5 X ULN (unless has Gilbert's disease) (to be performed within 2 days prior to start of cycle therapy) * CRITERIA TO PROCEED TO START OF CYCLE 1: AST =\< 2.5 x ULN (to be performed within 2 days prior to start of cycle therapy) * CRITERIA TO PROCEED TO START OF CYCLE 1: ALT =\< 2.5 x ULN (to be performed within 2 days prior to start of cycle therapy) * CRITERIA TO PROCEED TO START OF CYCLE 1: Serum creatinine \< 1.5 mg/dL or creatinine clearance of ≥ 60 mL/min per 24 hour urine test or the Cockcroft-Gault formula (to be performed within 2 days prior to start of cycle therapy) * CRITERIA TO PROCEED TO START OF CYCLE 1: If not receiving anticoagulants: International normalized ratio (INR) OR prothrombin (PT) ≤ 1.5 x ULN. If on anticoagulant therapy: PT must be within therapeutic range of intended use of anticoagulants (to be performed within 2 days prior to start of cycle therapy) * CRITERIA TO PROCEED TO START OF CYCLE 1: If not receiving anticoagulants: Activated partial thromboplastin Time (aPTT) ≤ 1.5 x ULN. If on anticoagulant therapy: aPTT must be within therapeutic range of intended use of anticoagulants (to be performed within 2 days prior to start of cycle therapy) * CRITERIA TO PROCEED TO START OF CYCLE 1: Left ventricular ejection fraction (LVEF) ≥ 50% (to be performed within 2 days prior to start of cycle therapy) * Note: To be performed before the first dose of LD chemotherapy * CRITERIA TO PROCEED TO START OF CYCLE 1: Corrected QT (QTc) ≤ 480 ms (to be performed within 2 days prior to start of cycle therapy) * CRITERIA TO PROCEED TO START OF CYCLE 1: O2 saturation \> 92% on room air (to be performed within 2 days prior to start of cycle therapy) * CRITERIA TO PROCEED TO START OF CYCLE 1: Women of childbearing potential (WOCBP): Negative urine or serum pregnancy test. If the urine test is positive or cannot be confirmed as negative, a serum pregnancy test will be required (to be performed within 2 days prior to start of cycle therapy) * CRITERIA TO PROCEED TO START OF CYCLE 1: Investigational drugs or devices within 30 days prior to start of cycle 1 therapy * NEXI-001 INCLUSION CRITERIA: Total bilirubin ≤ 2.5 X ULN (unless has Gilbert's disease) (to be performed within 1 days prior to NEXI-001 infusion) * NEXI-001 INCLUSION CRITERIA: AST =\< 3 x ULN (to be performed within 1 days prior to NEXI-001 infusion) * NEXI-001 INCLUSION CRITERIA: Serum creatinine \< 2 mg/dL (to be performed within 1 days prior to NEXI-001 infusion) * NEXI-001 INCLUSION CRITERIA: O2 saturation \> 92% on room air (to be performed within 1 days prior to NEXI-001 infusion) * CRITERIA TO PROCEED TO CYCLE 2: Patient has achieved a response of at least stable disease * CRITERIA TO PROCEED TO CYCLE 2: ANC must be at least 1,000/uL and platelets at least 50,000/uL to begin decitabine * CRITERIA TO PROCEED TO CYCLE 2: Patient has not experienced a ≥ grade 3 NEXI-001-related nonhematological AE that did not resolve to ≤ grade 2 within 72 hours Exclusion Criteria: * PARTICIPANT: Patients who have had 2 prior allogeneic (allo) HCTs * PARTICIPANT: Patients who have received more than 3 anti-leukemic treatments regimens since their allo HCT * PARTICIPANT: Vaccination with a live virus within six months prior to study treatment. * Inactivated influenza vaccination is allowed * PARTICIPANT: Active acute or chronic GVHD * PARTICIPANT: Known hypersensitivity to any component of the NEXI-001 T-cell product or fludarabine, cyclophosphamide, decitabine, or tocilizumab * PARTICIPANT: Clinically significant uncontrolled illness * PARTICIPANT: A second primary malignancy that has not been in remission for \> 2 years. Exceptions include the following resected lesions: * Non-melanoma skin cancer. * Carcinoma in situ. * Squamous intraepithelial lesions on Pap smear. * Localized prostate cancer (Gleason score \< 6). * Melanoma in situ * PARTICIPANT: Females only: Pregnant or breastfeeding * PARTICIPANT: Clinically significant cardiovascular disease: * Myocardial infarction or unstable angina within 6 months prior to the start of lymphodepleting (LD) chemotherapy. * Cerebral vascular accident or a transient ischemic attack within 6 months prior to the start of LD chemotherapy. * Clinically significant cardiac arrhythmia * Uncontrolled hypertension * Congestive heart failure (New York Heart Association Class III or IV) * Pericarditis or clinically significant pericardial effusion * Myocarditis * PARTICIPANT: History of autoimmune disease resulting in end organ injury or requiring systemic immunosuppression or systemic disease modifying therapy within 2 years prior to enrollment. * Patients with a history of autoimmune-related hypothyroidism on a stable dose of thyroid replacement hormone remain eligible. * Patients with controlled type 1 diabetes mellitus on a stable insulin regimen remain eligible * PARTICIPANT: Major trauma or major surgery within 4 weeks of enrollment * PARTICIPANT: Dementia or altered mental status that precludes understanding the informed consent form * PARTICIPANT: History of seizures or other chronic clinically significant neurologic disorders. * Patients with well-controlled seizures on anti-seizure medication without a seizure episode for ≥ 6 months remain eligible * PARTICIPANT: Any other condition that would, in the Investigator's judgment, contraindicate the patient's participation in the clinical study due to safety concerns with clinical study procedures * PARTICIPANT: Prospective participants who, in the opinion of the investigator, may not be able to comply with all study procedures (including compliance issues related to feasibility/logistics) * DONOR: The donor is pregnant or breastfeeding at the time of requested donation * DONOR: The donor had granulocyte colony stimulating factor (G-CSF) administered within one month, prior to leukapheresis * DONOR: The donor has an active bacterial or fungal infection, which is currently not responding to antimicrobial treatment * CRITERIA TO PROCEED TO START OF CYCLE 1: Other anti-leukemic (bridging) therapies within 14 days of start of cycle 1 therapy * CRITERIA TO PROCEED TO START OF CYCLE 1: Patients receiving systemic corticosteroid (\> 20 mg/day prednisone equivalent) or any other immunosuppressant agents at the time of initiation of LD chemotherapy. Intermittent topical, inhaled, or intranasal corticosteroids are allowed * CRITERIA TO PROCEED TO START OF CYCLE 1: Active acute or chronic GVHD (Note: must have resolved by the time of initiation of cycle 1 of therapy * CRITERIA TO PROCEED TO START OF CYCLE 1: Active or uncontrolled infection requiring antibiotics by the time LD chemotherapy is scheduled. Prophylactic and ongoing therapy for prior controlled infection is allowed * CRITERIA TO PROCEED TO START OF CYCLE 1: Females only: Pregnant or breastfeeding * CRITERIA TO PROCEED TO START OF CYCLE 1: Clinically significant cardiovascular disease: * Myocardial infarction or unstable angina within 6 months prior to the start of LD chemotherapy * Cerebral vascular accident or a transient ischemic attack within 6 months prior to the start of LD chemotherapy * Clinically significant cardiac arrhythmia * Uncontrolled hypertension * Congestive heart failure (New York Heart Association Class III or IV) * Pericarditis or clinically significant pericardial effusion * Myocarditis * History of seizures or other chronic clinically significant neurologic disorders. Patients with well-controlled seizures on anti-seizure medication without a seizure episode for ≥ 6 months remain eligible * CRITERIA TO PROCEED TO START OF CYCLE 1: Any other condition that would, in the Investigator's judgment, contraindicate the patient's participation in the clinical study due to safety concerns with clinical study procedures * NEXI-001 EXCLUSION CRITERIA: Patient has no evidence of NEXI-001-related nonhematological adverse events (AEs), e.g. cytokine release syndrome (CRS), immune effector cell-associated neurotoxicity syndrome (ICANS), hemophagocytic lymphohistiocytosis (HLH)/macrophage activation syndrome (MAS) * NEXI-001 EXCLUSION CRITERIA: Patient has not developed any of the exclusion criteria for treatment * CRITERIA TO PROCEED TO CYCLE 2: Patient has not developed any of the exclusion criteria for treatment