Donor Virus-Specific CMV or AdV CTL to Treat CMV or AdV Reactivation or Disease After Solid Organ or HCT

This trial studies the side effects and how well allogeneic cytomegalovirus-specific cytotoxic T lymphocytes (donor cytomegalovirus \[CMV\] specific cytotoxic T-lymphocytes \[CTLs\]) or allogeneic adenovirus-specific cytotoxic T lymphocytes (donor adenovirus-specific \[AdV\] specific CTLs) work in treating CMV or AdV reactivation or infection in participants who have undergone stem cell transplant or solid organ transplant. White blood cells from donors may be able to kill cancer cells in patients with cytomegalovirus or adenovirus that has come back after a stem cell or solid organ transplant.