Boosting transplants with donor immune cells to fight blood cancer relapse

Copied word for word from the study's registry entry, so the wording is the study team's rather than ours.

* INCLUSION CRITERIA: Inclusion Criteria - Recipient: * Histologically or cytologically confirmed hematologic malignancy classified as high or very high disease risk by the Refined Disease Risk Index for HCT including one of the following: * Acute myeloid leukemia (AML) with favorable cytogenetics (t(8;21), inv(16), t(15,17)) with induction failure (persistent disease without first achieving remission of any type) or active relapse * AML with intermediate cytogenetics (not classified as favorable or adverse) with induction failure or active relapse (AML with intermediate cytogenetics in morphologic complete remission \[CR\] with minimal residual disease detectable by any modality also will be eligible) * AML with adverse cytogenetics (complex karyotype with \>= 4 abnormalities) regardless of remission status * Low risk myelodysplastic syndrome (MDS) (\<= 5% blasts, including chronic myelomonocytic leukemia) with adverse cytogenetics (abnormal chromosome 7 or \>= 4 abnormalities) with induction failure or active relapse * High risk MDS (RAEB-1 or RAEB-2) with intermediate-risk cytogenetics (no abnormal chromosome 7 or \< 4 abnormalities) with induction failure or active relapse * High risk MDS (RAEB-1 or RAEB-2) with adverse cytogenetics (abnormal chromosome 7 or \< 4 abnormalities) regardless of remission status * Acute lymphoblastic leukemia (ALL) in CR \>= 2 or with induction failure or active relapse (ALL in CR1 with minimal residual disease detected also will be eligible) * Chronic myelocytic leukemia (CML) in blast crisis phase * Hodgkin lymphoma with stable or progressive disease * Mantle cell lymphoma with stable or progressive disease * Relapsed Burkitt lymphoma in CR or partial remission (PR) * Aggressive B-cell Non-Hodgkin Lymphoma (NHL) (e.g., diffuse large B-cell lymphoma, transformed indolent B-cell lymphoma) with stable or progressive disease * T-cell NHL with stable or progressive disease * Multiple myeloma (MM) with induction failure as defined by failure to achieve minimal response (CR, Very Good Partial Response \[VGPR\], or PR) or the development of progressive disease on primary therapy, or MM with active relapse as defined by previously treated myeloma that achieved a molecular response or better that then progressed * Age 18-65 years. * At least one potentially suitable HLA-haploidentical or HLA-matched donor * Karnofsky performance score \>60%, see Appendix A * Recipient participants must have adequate organ function as defined below: * Cardiac ejection fraction \>=45% by 2D ECHO; * Forced expiratory volume-1 (FEV-1), forced vital capacity (FVC), and diffusing capacity of the lung for carbon monoxide (DLCO) (corrected for hemoglobin) all of \>=50% predicted; * Estimated serum creatinine clearance of \>=60 ml/minute/1.73m\^2 calculated using eGFR in the clinical lab; * Total bilirubin \<=2X the upper limit of normal; * Alanine aminotransferase (ALT) and aspartate aminotransferase (AST) \<=3X the upper limit of normal. * Myeloablative conditioning is toxic to the developing human fetus and is teratogenic. For this reason, the following measures apply: --Women of child-bearing potential (WOCBP) and men must agree to use effective contraception (hormonal, surgical, IUD or barrier method of birth control; abstinence) from the start of conditioning regimen through at least one-year posttransplant. For the HLA haploidentical cohort the requirements are the same, however while taking sirolimus and for 12 weeks after the last dose, the birth control must be highly effective (hormonal, surgical, IUD, abstinence). * WOCBP must have a negative serum or urine pregnancy test within 7 days prior to enrollment. Inclusion Criteria - Donor: -Related donor (age \>=12) deemed suitable, eligible, and willing to donate, per clinical evaluations, who are additionally willing to donate blood, bone marrow, and stool for research. Related donors will be evaluated in accordance with existing Standard Policies and Procedures for determination of eligibility and suitability for clinical donation. EXCLUSION CRITERIA: Exclusion Criteria - Recipient: * Subjects who are receiving any other investigational agents. Prior experimental therapies must have been completed at least 3 weeks prior to the date of beginning conditioning. * Prior myeloablative conditioning for autologous or allogeneic HCT. * Active breastfeeding. * Active malignancy of non-hematopoietic type (excluding non-melanoma skin cancers) which is metastatic, relapsed/refractory to treatment, or locally advanced and not amenable to curative treatment. This excludes non-melanoma skin cancers. * Uncontrolled intercurrent illness (e.g. severe endocrinopathy, disseminated intravascular coagulation, profound electrolyte disturbance, active hepatitis, uncontrolled dental infection) that in the opinion of the PI would make it unsafe to proceed with transplantation. Exclusion Criteria - Donor: None.