Non-manageable graft versus host disease. Active graft-versus-host disease. Active graft-versus-host disease. Participants must have cGVHD (as defined by the National Institutes of Health Consensus Development Project on Criteria for Clinical Trials in Chronic Graft-Versus-Host Disease) Current or history of graft versus host disease Active chronic graft versus host disease requiring immunosuppressive treatment. Patients with >= grade 3 acute graft-versus-host disease are excluded from the study Patients with moderate or severe chronic graft-versus host requiring more than 20 mg of oral prednisone or equivalent therapy are excluded from the study Patients who have undergone allogeneic stem cell transplant > 12 months, without active graft-versus-host-disease, and not on immunosuppression for prevention of graft-versus-host disease are eligible there are no signs or symptoms of graft versus host disease, other than Grade 1 skin involvement. GRAFT CRITERIA: No prior therapy with mTOR inhibitors except for rapalog treatment as part of graft-versus-host (GVH) prophylaxis or treatment Graft must be a CD3+ T-cell replete PBSC graft or non-manipulated BM graft. Presence of chronic graft versus host disease (cGVHD) in an organ other than lung Active graft-versus-host disease. Active hepatic graft-versus-host disease Active graft versus-host disease and must not be on immunosuppression Participants who have received immunosuppressive therapy for graft versus host disease within 2 weeks prior to randomization Any level of acute graft versus host disease GRAFT CRITERIA: DONOR: Donors will be selected to minimize HLA mismatch in the host-versus-graft direction Active acute graft-versus-host disease of any grade or chronic graft-versus-host disease of Grade 2 or higher Less than 3 months since received hematopoietic stem cell transplantation, autologous or allogeneic and/or have clinically significant graft versus-host disease requiring treatment at the time of screening and/or patient having a history of severe graft versus-host disease; Signs of or symptoms of active graft versus host disease Patients with symptomatic oral chronic graft-versus-host disease (sensitivity score >= 4) Graft-versus-host disease criteria: Active graft-versus-host disease. Active graft-versus-host disease. Active acute grade III-IV graft-versus-host disease Ongoing graft-versus-host disease Prior use of fludarabine, as this agent has been associated with higher subsequent rates of graft versus host disease Active graft?versus?host disease requiring systemic treatment Patients who have undergone allogeneic stem cell transplant within 12 months, without active graft-versus-host-disease, and not on immunosuppression for prevention of graft-versus-host disease are eligible Subjects who have clinically significant graft versus host disease requiring treatment and /or have >grade 2 persistent non hematological toxicity related to transplant Participants who received prior allogenic transplant must have had no active graft-versus-host disease. Active graft versus host disease. Presence of uncontrolled graft-versus-host disease Evidence of graft versus host disease > grade II Evidence of chronic or acute graft versus host disease or on-going treatment for graft versus host disease from prior allogeneic stem cell transplantation Current immunosuppressive treatment for graft versus host disease Active graft versus host disease No signs or symptoms of graft versus host disease other than Grade 1 skin involvement Has clinically significant graft-versus-host disease requiring treatment Ongoing graft-versus (vs)-host disease Evidence of uncontrolled graft-versus-host disease Receiving systemic therapy for chronic graft-versus-host disease (topical therapy is allowed) Post allo patients must not have active graft versus-host disease and be off all immune suppression (other than steroids, as above) Any evidence of cGVHD or late acute graft versus host disease (aGVHD) between enrollment and first dose of obinutuzumab Active systemic treatment for graft versus host disease. HSCT recipients with active and/or chronic graft versus host disease Subject has received a prior HCT and has residual Chronic Graft-versus-host Disease (cGVHD) Hematopoietic stem cell transplantation (HSCT) within 60 days of screening, or receipt of immunosuppressive therapy for graft-versus-host disease treatment or prophylaxis within 14 days of first protocol therapy, or active graft-versus-host-disease Active graft-versus-host disease.