Reduced Intensity Chemotherapy and Total Body Irradiation before TCR-alpha/beta+ T-lymphocytes Donor Transplant in Treating Participants with High-Risk Myeloid Diseases

Inclusion Criteria

• Patients with one of the high risk myeloid diseases as outlined below. Patients must have =< 5% blasts on the last bone marrow (BM) evaluation prior to starting the conditioning regimen. Diseases included on this protocol include: * Acute myeloid leukemia (AML) in first complete remission (CR1) with intermediate or high risk features as defined below: ** Cytogenetic abnormalities which are not considered “good risk” cytogenetic features (i.e. t(8:21), t(15:17), inv 16 without c-kit mutations. And/or ** Therapy related AML with history of antineoplastic therapy (radiation and/or chemotherapy) And/or ** Normal karyotype with mutations of FLT3, RUNX1, TP53 mutation, ASXL1 or any others that are considered to be high risk * AML in >= 2nd remission * Myelodysplastic syndrome, myeloproliferative neoplasms, or MDS/myeloproliferative neoplasms (MPN) overlap syndrome with: ** International prognostic scoring system risk score intermediate-2 (INT-2) or high risk at the time of transplant evaluation. And/or ** Any risk category if life-threatening cytopenia exists And/or ** Karyotype or genomic changes that indicate high risk for progression to acute myelogenous leukemia, including abnormalities of chromosome 7 or 3, mutations of TP53, or complex or monosomal karyotype. * Chronic myelomonocytic leukemia (CMML). * Chronic myeloid leukemia (CML) with the following features: ** Patients who have failed or are intolerant to BCR-ABL tyrosine kinase inhibitors. And/or ** CML with BCR-ABL mutation consistent with poor response to tyrosine kinase inhibition (e.g. T351l mutation) * Patients with severe aplastic anemia.
• Chronic lymphocytic leukemia (CLL) with high risk disease as defined by the European Society for Blood and Marrow Transplantation (EBMT) consensus criteria
• Non-Hodgkin lymphoma meeting both of the following criteria: * Responding to therapy prior to enrollment. * Relapse after prior autologous bone marrow transplant or are ineligible for autologous bone marrow transplant.
• Multiple Myeloma with disease in the following categories: * Patients with relapsed multiple myeloma following autologous stem cell transplantation who have achieved at least partial response following additional chemotherapy * Patients with high risk cytogenetics at diagnosis must have achieved at least a partial response following autologous stem cell transplantation. Patients must have complex karyotype, del17p, t4;14, and/or t14;16 by fluorescence in situ hybridization (FISH) and/or del13 by karyotyping.
• Each patient must be willing to participate as a research participant and must sign an informed consent form.
• Patients be >= 18 years old
• Patients must have a Karnofsky (adult) or performance status >= 70%.
• Cardiac: Asymptomatic or if symptomatic, then left ventricular ejection fraction (LVEF) at rest must be >= 40% and must improve with exercise.
• Hepatic: < 5 x upper limit of normal (ULN) alanine aminotransferase (ALT).
• Hepatic: < 2 x ULN total serum bilirubin, unless there is congenital benign hyperbilirubinemia.
• Renal: Creatinine clearance (CrCl) >= 30 ml/min (measured or calculated/estimated).
• Pulmonary: Asymptomatic or if symptomatic, diffusion capacity of the lung for carbon monoxide (DLCO) > 50% of predicted (corrected for hemoglobin).
• DONOR: Must be a 10/10 human leukocyte antigen (HLA) genotypically match related or unrelated donor at all A, B, C, DRB1, and DQB1 loci, as tested by deoxyribonucleic acid (DNA) analysis.
• DONOR: Able to provide informed consent for the donation process per institutional standards.
• DONOR: Meet standard criteria for donor collection as defined by the National Marrow Donor Program Guidelines.

Exclusion Criteria

• Prior allogenic hematopoietic stem cell transplantation.
• Prior radiation therapy with 400 cGY or more of TBI.
• BM with increased fibrosis (reticulin stain > 1/3).
• Active and uncontrolled infection at time of transplantation.
• Human immunodeficiency virus (HIV) infection.
• Seropositivity for human T-lymphotrophic virus (HTLV)-1.
• Inadequate performance status/organ function.
• Pregnancy or breast feeding.
• Patient or guardian unable to give informed consent or unable to comply with the treatment protocol including appropriate supportive care, follow-up, and research tests.