Rivogenlecleucel Donor Lymphocyte Immunotherapy in Treating Patients With Recurrent Blood Cancers After Stem Cell Transplant

Inclusion Criteria: * Original HCT donor: The most recent transplant was from a related or unrelated donor with an allele-level match to the recipient at HLA-A, HLA-B, HLA-C, HLA-DRB1, HLA-DQB1 (10/10) or with a single antigen or allele mismatch (9/10), or with one or two umbilical cord blood units that are individually antigen-level matched to the recipient in at least 4 of 6 HLA loci (HLA-A, -B, -DRB1). * Graft source: The most recent transplant (and any previous allotransplant) was from mobilized peripheral blood stem cells and/or bone marrow and/or umbilical cord blood. * Transplant number: The relapse occurred after a 1st, 2nd or 3rd allogeneic HCT. * Diseases: The subject relapsed or progressed with the same or related disease which prompted the previous allogeneic HCT. * Leukemia (acute myeloid leukemia \[AML\], acute lymphoblastic leukemia \[ALL\], chronic myelogenous leukemia \[CML\], chronic lymphocytic leukemia \[CLL\], chronic myelomonocytic leukemia \[CMML\], blastic plasmacytoid dendritic cell, biphenotypic, bilineage) * Myelodysplasia * Myeloproliferative neoplasm * Myelodysplasia/myeloproliferative neoplasms (including "unclassified"). * For subjects with persistent or relapsed acute B-cell lymphoblastic leukemia, the subject must have previously received a chimeric antigen receptor (CAR) T cell product and blinatumomab at any point (not necessarily to treat the current relapse or after the current allogeneic transplant), unless the subject is ineligible for a commercially-available CAR T cell product or blinatumomab or declines those treatments. * Relapse stage: The disease stage may be ascertained any time after engraftment of the original donor cells, where engraftment is defined as the first of 3 consecutive days of absolute neutrophil count \>= 500/uL unsupported by granulocyte-colony stimulating factor (G-CSF) within the preceding 5 days. (Note that eligibility for the protocol does not require an ANC \> 500/uL at the time of enrollment.) Subjects with any of the following disease stages are eligible for this protocol: * Morphologic disease persistence or relapse (defined as \>= 5% bone marrow blasts in a bone marrow aspirate or biopsy, and/or the presence of circulating blasts detected by automated differential or review of the peripheral blood film and confirmed to be malignant, and/or \>= 5% bone marrow blasts with aberrant immunophenotype consistent with malignancy detected by flow cytometry of blood or marrow) * Extramedullary disease persistence or relapse (e.g., granulocytic sarcoma, leukemia cutis) * A new hematologic malignancy related to the prior transplant indication (such as AML arising from myelodysplastic syndrome \[MDS\] or ALL arising from mixed lineage leukemia \[MLL\]-rearranged AML) as per above * Subject may have received prior DLI and/or prior genetically-modified DLI such as CAR-T cells and/or prior chemotherapy other than purine analogues to treat minimal residual disease (MRD) or morphologic/clinical relapse or disease persistence but must have had relapse as defined at some point after the most recent HCT to be enrolled on this study. * Adequate HCT donor chimerism as indicated by original-donor T cell chimerism (from the most recent HCT) in peripheral blood \>= 50%. In the case of double umbilical cord transplant, one of the cord donors should constitute \>= 50% of the peripheral blood T cells. * For subjects who received genetically-modified T cells such as CAR-T cells or antigen-specific T-cell receptor (TCR) T cells, the non-engineered, polyclonal T cells must constitute at least 50% of total peripheral blood T cells. * Performance status: Karnofsky or Lansky performance status (PS) \>= 80% or Eastern Cooperative Oncology Group (ECOG) PS 0-1. * Capable of giving informed consent, if the subject is 18 years of age or older. A parent or legal representative will be asked to consent for patients young than 18 years old. * DONOR: The donor must be available to undergo apheresis at the Seattle Cancer Care Alliance, South Lake Union site or at Bloodworks Northwest. * DONOR: The donor must be suitable for medical reasons to donate according to institutional Standard Practice Guidelines and National Marrow Donor Program (NMDP) guidelines, including the absence of transmissible infections (hepatitis A, B, HIV, HTLV I/II, syphilis, West Nile virus \[WNV\] and Chagas disease). * DONOR: The donor must not be homozygous to the recipient at an HLA allele that is common to recipient and donor. (Homozygosity at antigenic resolution is permitted.) Therefore, people who are homozygous at any HLA allele will be excluded. * DONOR: The donor may not knowingly be related to a subject on this study by blood (e.g., sibling, parent, child, cousin). This is to prevent "coercive recruitment" of donors. * DONOR: The donor must be 18-50 years old. * DONOR: Capable of giving informed consent. * DONOR: Not pregnant or breastfeeding at the time of apheresis. * DONOR: Not on any prescription medications other than birth control pills or vitamins. * DONOR: The donor must be cytomegalovirus (CMV) seronegative at the time of screening. For the product to be banked, the CMV polymerase chain reaction (PCR) done on the day of apheresis must be negative. Exclusion Criteria: * Subjects who currently have \>= grade 1 acute GVHD or any chronic GVHD manifestation beyond pre-existing ocular or oral sicca or pre-existing sclerosis. * Subjects with oral or ocular sicca attributable to prior chronic GVHD should be excluded if there are other signs of active GVHD such as eosinophilia not explained by other causes (e.g. recurrent malignancy, medication). * Inability to have their dose of corticosteroids tapered to =\< 0.25 mg/kg/day prednisone-equivalent for a minimum of 7 days prior to BPX-501 cell infusion. This determination will be made by the subject's treating physician (who is managing the subject's GVHD), documented in a clinical note, and verified by the protocol principal investigator (Pl). * Inability to stop all non-steroid immunosuppressive drugs (excepting beclomethasone, budesonide, topical skin corticosteroids, topical tacrolimus, Restasis eye drops, or low doses of glucocorticoids and/or mineralocorticoids given for adrenal insufficiency) for at least 2 weeks without development of grade 1 or greater acute GVHD according to the modified Keystone and NIH GVHD Morbidity Scales. * The use of purine analogue chemotherapy (including pentostatin), alemtuzumab, anti-thymocyte globulin, anti-lymphocyte globulin, total body irradiation, and total lymphoid irradiation at any time after infusion of the index HCT graft. All other debulking protocols and patients who received any form of ex-vivo T-cell depleted graft should be cleared by the protocol PI before confirming subject eligibility for this trial. * Central nervous system (CNS) leukemia (including leukemia detectable in the cerebrospinal fluid and/or solid chloromas and/or leptomeningeal leukemia) that has not cleared with radiation, intrathecal or systemic therapy. * Inadequate cardiac function defined by left ventricular ejection fraction (LVEF) \< 40%. * Inadequate pulmonary function defined by any one of the following: forced expiratory volume in one second (FEV1), forced vital capacity (FVC), or corrected diffusion capacity of the lung for carbon monoxide (DLCO) \< 40% predicted, or peripheral capillary oxygen saturation (SpO2) \< 92% on room air. * Direct bilirubin \> 3 x upper limit of normal. * Aspartate aminotransferase (AST)/alanine aminotransferase (ALT) \> 5 X upper limit of normal. * Creatinine \> 2 X upper limit of normal for age or on dialysis. * Concurrent active malignancy other than the malignancy under treatment with BPX-501. * Uncontrolled infection, including no invasive fungal infection during the acute antifungal treatment phase (secondary prophylaxis for fungal infection is permitted) and no viral respiratory tract infection that has not cleared as evidenced by a negative nasopharyngeal aspirate or nasopharyngeal swab. * Positive for human immunodeficiency virus (HIV), chronic hepatitis B, chronic hepatitis C, or human T-lymphotropic virus (HTLV) pre-transplant (To continue on study, this testing also needs to be confirmed negative within 30 days pre-DLI). * Fertile men and women unwilling to use contraceptives before, during, and for 4 months after BPX-501 infusion. * Pregnancy: For females of childbearing potential, a pregnancy test is required at enrollment and must be confirmed negative within 7 days before BPX-501 T cell infusion. * Medical or psychological conditions present within 30 days prior to enrollment that would make the subject unsuitable for cell therapy or for research trial participation at the discretion of the PI.