Donor Regulatory T Cells in Treating Patients With Visceral Acute Graft-versus-Host Disease After Stem Cell Transplant

Inclusion Criteria: * Visceral aGVHD defined as: at least stage III/IV acute liver or stage II/III gastrointestinal (GI) GVHD by clinical criteria and/or GI and/or liver biopsy confirmation showing no alternative explanation for symptoms of GVHD * Ability to understand and willingness to sign a written informed consent form * Must have a 7/8 or 8/8 or haploidentical related donor matched at the human leukocyte antigen (HLA)-A, B, C, DRB1 who was evaluated and provided the donor transplant graft * Myeloablative or non-myeloablative allogeneic hematopoietic cell transplantation * Karnofsky performance status \>= 50 * DONOR: Age \>= 18 to =\< 77 years old * DONOR: Karnofsky performance status of \>= 70% defined by institutional standards * DONOR: Must be the same sibling donor from whom the recipient's blood and marrow graft was collected for the original allogeneic transplant that is HLA 7/8 or 8/8 or haploidentical matched at the HLA-A, B, C, and DRB1 * DONOR: Serologies for human immunodeficiency virus (HIV) antigen (Ag), HIV 1 and HIV 2 antibody (Ab), human T-cell lymphotropic virus (HTLV) 1 and HTLV 2 Ab, hepatitis B surface antigen (sAg) or polymerase chain reaction (PCR)+, or hepatitis C Ab or PCR+, syphilis (Treponema) screen and HIV 1 and hepatitis C by NAT (nucleic acid testing) have been collected prior to apheresis * DONOR: Female donors of child-bearing potential must have a negative serum or urine beta-human chorionic gonadotropin (HCG) test within two weeks of apheresis * DONOR: Capable of undergoing leukapheresis, have adequate venous access, and be willing to undergo insertion of a central catheter should leukapheresis via peripheral vein be inadequate * DONOR: Donor selection will be in compliance with 21 Code of Federal Regulations (CFR) 1271 Exclusion Criteria: * Uncontrolled infections not responsive to antimicrobial therapy requiring intensive critical care * Progressive malignant disease, including post-transplant lymphoproliferative disease unresponsive to therapy * Cytomegalovirus colitis or enteritis as defined by cytomegalovirus (CMV) shell vial or culture positivity from endoscopic biopsy the discretion of the treating physician based upon PCR positivity, clinical presentation and histology * Respiratory insufficiency with oxygen requirement \> 4 L nasal cannula * Multi-organ failure * DONOR: Evidence of active infection or viral hepatitis * DONOR: HIV positive * DONOR: Pregnant donor * DONOR: Factors which place the donor at increased risk for complications from leukapheresis