Donor Regulatory T-cells for cGVHD in Patients Who do Not Obtain Complete Remission With Ruxolitinib

Inclusion Criteria: * Recipient of allogeneic hematopoietic stem cell transplantation * Participants must have steroid-refractory cGVHD and had obtained any response other than progression after at least 12 weeks of treatment with ruxolitinib. Steroid-refractory cGVHD is defined as having persistent signs and symptoms of cGVHD despite the use of prednisone at ≥ 0.25 mg/kg/day (or 0.5 mg/kg every other day) for at least 4 weeks (or equivalent dosing of alternate glucocorticoids) without complete resolution of signs and symptoms. * Stable dose of glucocorticoids for 4 weeks prior to enrollment. * No addition or subtraction of other immunosuppressive medications (e.g., calcineurin-inhibitors, sirolimus, mycophenolate-mofetil) for 4 weeks prior to enrollment. The dose of immunosuppressive medicines may be adjusted based on the therapeutic range of that drug. * No age limit. In the case of children participating in the study, the informed consent will be signed by a parents or legal guardians. * Eastern Cooperative Oncology Group scale performance status 0-2 * Participants must have adequate organ function * Women of child-bearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry and for the duration of study participation. Should a woman become pregnant or suspect she is pregnant while participating in this study, she should inform her treating physician immediately. * Ability to understand and the willingness to sign a written informed consent document Exclusion Criteria: * Ongoing prednisone requirement \>1 mg/kg/day (or equivalent). * Concurrent use of calcineurin-inhibitor plus sirolimus (either agent alone is acceptable). * History of active thrombotic microangiopathy, hemolytic-uremic syndrome or thrombotic thrombocytopenic purpura in the last 6 months. * New immunosuppressive medication in the 4 weeks prior to enrollment. * Extra-corporeal Photopheresis or rituximab therapy in the 4 weeks prior to enrollment. * Post-transplant exposure to T-cell or interleukin-2 targeted medication within 100 days prior to enrollment. * Donor lymphocyte infusion within 100 days prior to enrollment. * Active malignant relapse. * Active uncontrolled infection. * Organ transplant (allograft) recipient. * HIV-positive individuals on combination antiretroviral therapy are ineligible. * Individuals with active uncontrolled hepatitis B or C are ineligible as they are at high risk of lethal treatment-related hepatotoxicity after hematopoietic stem cell transplant. * Other investigational drugs within 4 weeks prior to enrollment, unless cleared by the Principal Investigator. * Pregnant women are excluded from this study.