Clinically suspected Grade II to IV aGVHD as per MAGIC criteria, occurring after allo-HSCT and any GVHD prophylaxis regimen. Has received more than 1 allo-HSCT. Participants with evidence of relapsed primary disease, or participants who have been treated for relapse after the allo-HSCT was performed. Patients who received Allo-Stem cell Transplantation(Allo-SCT) within 12 months. Completed allogeneic stem cell transplant (allo-SCT) or are eligible for and willing to complete allo-SCT Any patients eligible for allogeneic stem cell transplantation (allo-SCT) and willing to undergo allo-SCT as determined at time of screening for trial; patients who are ineligible or not interested in undergoing allo-SCT will be eligible for the trial Patients who are receiving any other investigational agents, with the exception of virus-specific cytotoxic T-cells for the treatment of viral infection/reactivation prior to allo BMT PHASE I: For the post allo-HCT cohort, chimerism analysis must demonstrate > 95% either donor or recipient in peripheral blood CD3 cells, must be done within 1 month of enrollment (no mixed chimerism are eligible) Patients with Allo-SCT on active GVHD or immunosuppression therapy within 3 months prior to C1D1. Acceptable allogeneic stem cell donor with imminent plans to proceed with allo-SCT. Allo-HSCT within 90 days of leukapheresis Prior allo-HCT less than three months from the time of enrollment Previous allo-HSCT of any kind Allowed prior allogeneic hematopoietic stem cell transplantation (allo-HCT) regardless of stem cell source; patients must be at least 3 months post allo-HCT (at time of treatment start); mismatched transplantations would be allowed If post allo-HCT, then patient must have baseline donor T cell chimerism of >= 20% (from peripheral blood); evaluation can be made within 4 weeks of treatment start Acute Leukemia (AML or ALL) patient or allo-HCT recipient with a diagnosis of IA Prior allogeneic (allo)-hematopoietic cell transplantation (HCT) less than three months from the time of enrollment Previous allogeneic (allo)-transplant of any kind Has received more than 1 allo-HSCT. Subjects with evidence of relapsed primary disease, or subjects who have been treated for relapse after the allo-HSCT was performed. Unresolved toxicity or complications (other than acute GVHD) due to previous allo-HSCT. Received Janus kinase inhibitor therapy after allo-HSCT for any indication. Treatment with a JAK inhibitor before allo-HSCT is permitted. Inadequate recovery from toxicity and/or complications from the prior allo-HSCT. Patients underwent allo-SCT with intermediate risk and high risk AML and high risk MDS (defined by American Society for Blood and Marrow Transplantation [ASBMT] criteria), who are within 60 to 100 days after allo-SCT Prior AP, BC or allo-transplant Recipient of 1 allo-HSCT but not more than 1 allo-HSCT. Relapse of underlying malignant disease after allo-HSCT. For Pre-allo Part A (before stem cell transplant): Relapsed or refractory AML (greater than 5% blasts) For Post-allo Part B: Treatment must begin at least 60 days, but no more than 100 days post-transplant. For Pre-allo Part A (before stem cell transplant): Prior alloSCT For Post-allo Part B: Active GVHD Grade 2 or higher For Post-allo Part B:History of veno-occlusive disease requiring defibrotide For Post-allo Part B: History of Grade 2 or higher hepatic GVHD A patient is eligible for second enrollment (allo-cellular therapy) if all of the following inclusion criteria are met: At least a partial remission before allo-SCT Patients undergoing their first T-cell replete allogenic (allo)-HCT for CLL, MCL, follicular lymphoma (FL), Hodgkin disease (HD) Patient undergoing an allo-SCT Planned to undergo allo-HSCT