Phase 1 Infused Donor T Regulatory Cells in Steroid Dependent/Refractory Chronic GVHD

Inclusion Criteria: * Steroid dependent/refractory cGVHD defined as: * Steroid dependent disease: Persistent cGVHD manifestations requiring a glucocorticoid dose \>= prednisone 0.25 mg/kg/day (0.5 mg/kg orally \[po\] every other day) for at least 12 weeks * Steroid refractory disease: Progressive cGVHD manifestations despite treatment with a glucocorticoid dose \>= prednisone 0.5 mg/kg/day (1 mg/kg po every other day) for at least 4 weeks * Participants must be receiving systemic glucocorticoid therapy for cGVHD; all immunosuppressive therapy may include but not be limited to tacrolimus, sirolimus, CellCept, cyclosporine, and systemic corticosteroid must be at stable doses for 28 days prior to the first cell infusion * Chronic GVHD manifestations that can be followed on physical or laboratory exam; these include but are not necessarily limited to: * Skin changes * Oral mucosa changes * Bronchiolitis obliterans * Ocular changes * Karnofsky performance status \>= 60 * Serum creatinine =\< 2 mg/dL * Absolute neutrophil count (ANC) \> 1 x 10\^9/L * Aspartate aminotransferase (AST) or alanine aminotransferase (ALT) =\< 20 x upper limit of normal (ULN) or * Total bilirubin =\< 10 x ULN * Allogeneic hematopoietic cell transplant recipient * Transfusion independent * Oxygen saturation during exertion is maintained at \>= 88% on room air * Does not have clinically significant, symptomatic uncontrolled heart disease (e.g., unstable angina, congestive heart failure, or uncontrolled hypertension) * DONOR: Age \>= 18 to =\< 75 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 human leukocyte antigen (HLA) 7/8 or 8/8 matched at the HLA-A, B,C, DRB1 * DONOR: Serologies for human immunodeficiency virus (HIV) antigen (Ag), HIV 1 and HIV 2 antibody (Ab), human T-lymphotropic virus type I (HTLV 1) and HTLV 2 Ab, hepatitis B surface antigen (sAg) or polymerase chain reaction positive (PCR+), or hepatitis C Ab or PCR+, Syphilis (Treponema) screen and HIV 1 and hepatitis C by nucleic acid testing (NAT) 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 three 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: * Original transplant utilized an unrelated donor graft * Uncontrolled infections that are not responsive to antimicrobial therapy * Progressive malignant disease, including post-transplant lymphoproliferative disease unresponsive to therapy * Second malignancy except for skin cancer within the last 5 years * Received any investigational agent =\< 28 days before Treg infusions * Received filgrastim (GCSF) treatment within one month of enrollment * Received a donor lymphocyte infusion (DLI) or hematopoietic cell transplantation (HCT) within 3 months of enrollment * DONOR: Evidence of active infection or viral hepatitis * DONOR: HIV positive * DONOR: Pregnant donor