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