Stem cell infusions (with or without total body irradiation [TBI]):\r\n* Allogeneic (non-autologous) bone marrow or stem cell transplant, or any stem cell infusion including donor lymphocyte infusion (DLI) or boost infusion: >= days after infusion, and no evidence of graft-versus-host disease (GVHD)\r\n* Autologous stem cell infusion including boost infusion: >= days Participants must be at least days after the transplantation or a donor lymphocyte infusion Donor lymphocyte infusion within days prior to enrollment Planned use of prophylactic donor lymphocyte infusion (DLI) therapy. Steroid refractory cGvHD occurring after a non-scheduled donor lymphocyte infusion (DLI) administered for preemptive treatment of malignancy recurrence. Participants who have received a scheduled DLI as part of their transplant procedure and not for management of malignancy relapse are eligible Patients receiving a donor lymphocyte infusion within weeks of planned T cell infusion Patients receiving a donor lymphocyte infusion within weeks of planned T cell infusion Subjects who developed aGVHD after unplanned donor lymphocyte infusion. TREATMENT WITH SJCAR: Prior to planned SJCAR infusion, patients with a history of prior allogeneic HCT must be at least months from HCT, have no evidence of active GVHD and have not received a donor lymphocyte infusion (DLI) within the days prior to planned infusion Patients receiving prednisone > . mg/kg/day or equivalent at time of enrollment, or who have received anti- thymocyte globulin (ATG) within days or have received donor lymphocyte infusion (DLI) or Campath within days of enrollment. Donor lymphocyte infusion (DLI) within days prior to enrollment Subjects must be at least days from prior stem cell transplant (autologous or allogeneic) or donor lymphocyte infusion (DLI) Donor lymphocyte infusion administered to treat relapse or loss of donor chimerism Prior donor lymphocyte infusions (DLIs) are not necessary Donor lymphocyte infusions (DLI) within weeks of JCAR administration Patients who have had donor lymphocyte infusion (DLI) within weeks prior to registration Donor lymphocyte infusion within days prior to enrollment Subjects must not have received a donor lymphocyte infusion (DLI) within weeks prior to the first dose of study medication Has received donor lymphocyte infusion (DLI) product within weeks of CAR T cell infusion Patients receiving prednisone > . mg/kg/day at time of enrollment, or have received anti-thymocyte globulin (ATG) within days or have received donor lymphocyte infusion (DLI) or Campath within days of enrollment ELIGIBILITY CRITERIA FOR T-CELL PRODUCT INFUSION: Absolute lymphocyte count < /ul or patient has received lymphodepleting chemotherapy administered at least hours prior to T cell infusion Patients who have received donor lymphocyte infusion (DLI) within days of Viralym-A infusion. Patients receiving prednisone > . mg/kg/day at time of enrollment, or have received anti-thymocyte globulin (ATG), donor lymphocyte infusion (DLI) or Campath within days of enrollment Patients who have received donor lymphocyte infusion (DLI) within days Patients who have received donor lymphocyte infusion (DLI) within days No change in dosing of immunosuppressive agents in the weeks preceding the naive T-cell depleted donor lymphocyte infusion A commitment not to electively taper for a minimum of days, the immunosuppressive medications ongoing at time of naive T-cell depleted donor lymphocyte infusion Patients developing SR aGvHD after donor lymphocyte infusion (DLI) or after withdrawal of immunosuppression are eligible Donor lymphocyte infusion within weeks prior to treatment start if post-transplant Logistical requirements:\r\n* Space available in infusion room\r\n* Outpatient infusion pump available if continuous infusion required\r\n* Case discussed with infusion room nursing staff Allogeneic (non-autologous) bone marrow or stem cell transplant, or any stem cell infusion including donor lymphocyte infusion or boost infusion: ? days after infusion and no evidence of graft versus host disease (GVHD) Autologous stem cell infusion including boost infusion: ? days Unplanned donor lymphocyte infusion (DLI) for residual or relapsed malignancy or mixed chimerism. DLI as part of the planned HCT protocol is allowed. Less than months following bone marrow or peripheral blood stem cell transplantation or treatment with donor lymphocyte infusion (DLI). Donor lymphocyte infusion (DLI) is considered a reinduction attempt. At least weeks (from first dose) elapsed from donor lymphocyte infusion (DLI) without conditioning. Patients with acute GVHD developing after a donor lymphocyte infusion. Allogeneic cellular therapy: Any donor lymphocyte infusions (DLI) must be completed > weeks prior to CTL infusion Allogeneic cellular therapy: Any donor lymphocyte infusions (DLI) must be completed > weeks prior to CTL infusion The patient has, or accepts to have, an acceptable infusion device for infusion of melflufen Donor lymphocyte infusion (DLI) within days prior to enrollment Donor lymphocyte infusion within days prior to enrollment Participant with post-transplant exposure to donor lymphocyte infusion (DLI), or T-cell or IL- targeted medication (e.g. anti-thymocyte globulin [ATG], alemtuzumab, basiliximab, denileukin diftitox) within days prior Donor lymphocyte infusion within wks prior to starting dose of AMG . Donor lymphocyte infusions (DLI) within months prior to leukapheresis Stem cell infusions (with or without total body irradiation): Allogeneic (non-autologous) bone marrow or stem cell transplant, or any stem cell infusion including donor leukocytes infusion or boost infusion: ? days after infusion and no evidence of graft versus host disease; Autologous stem cell infusion including boost infusion: ? days Patients may have had prior treatment for myelodysplastic syndrome (MDS) or acute myelogenous leukemia (AML), including lenalidomide; patient may have had prior autologous or allogeneic transplant (family member, unrelated donor, or cord blood) if there is at least days between transplant and study entry; patients may also have had donor lymphocyte infusion if there is at least days between donor lymphocyte infusion and study entry; patients on immunosuppression are also eligible Subjects who received an unplanned (not part of the original transplant therapy plan) donor lymphocyte infusion. Allogeneic cellular therapy: Donor lymphocyte infusions (DLI) are prohibited within weeks prior to leukapheresis Participant whose GvHD developed after donor lymphocyte infusion The patient will need to be available for evaluation within hours of symptoms of GVHD, occurring within days of the planned donor lymphocyte infusion No chemotherapy, RT, donor lymphocyte infusion (DLI) or biologic therapy for lymphoma at least weeks prior to scheduled treatment Patients who have received donor lymphocyte infusion (DLI) within days of Viralym-C infusion. Received donor lymphocyte infusion (DLI) within days Treatment with Epstein-Barr virus cytotoxic T lymphocytes, chimeric antigen receptor (CAR)-T cells directed against B cells, or unselected donor lymphocyte infusion (DLI) within weeks of Cycle Day Patients may also have had donor lymphocyte infusion if there is at least days between donor lymphocyte infusion and study entry Anticipated planned donor lymphocyte infusion in the first months post-SCT Patients may have received a prior allogeneic hematopoietic stem cell transplant (alloHSCT) for any indication and from any donor; patients developing cGvHD after donor lymphocyte infusion (DLI) are also eligible Received a donor lymphocyte infusion (DLI) or hematopoietic cell transplantation (HCT) within months of enrollment Planned use of prophylactic donor lymphocyte infusion (DLI) therapy Donor lymphocyte infusion within days prior Patients who have received donor lymphocyte infusion (DLI) within days Patients whose GVHD developed after donor lymphocyte infusion (DLI) Donor lymphocyte infusion in the preceding days Subjects who have received stem cell boost or delayed donor lymphocyte infusion within days of enrollment, including day of enrollment FOR THE SUBJECTS ENROLLED IN YEAR : Subjects who have received stem cell boost or delayed donor lymphocyte infusion within days of enrollment, including day of enrollment Subjects who have received stem cell boost or delayed donor lymphocyte infusion within days of enrollment, including day of enrollment No donor lymphocyte infusion (DLI) prior to day , and no plans for a DLI in the upcoming days Patients may have received no more than one donor lymphocyte infusion (DLI), DLI must have been administered > weeks prior to enrollment on study, and no plans for a DLI in the upcoming days Donor lymphocyte infusions (DLI) within weeks of JCAR administration At least weeks (from first dose) has elapsed from donor lymphocyte infusion (DLI) without conditioning.