This is termed allogenic recognition. It may occur by direct and indirect routes.
In the direct route the donors allogenic antigen presenting cell presents with an antigen bound to donor MHC. The recipient's T cell then thinks that the donor MHC is self MHC presenting with a pathogenic antigen. The T cell then mounts an immunological response to the donor MHC. The repose of the T cell depends on whether it is a CD4+ T cell or a CD8+ T cell. If it is a CD4+ T cell (T-helper), it will release cytokines to promote the immune response. B cells are activated to manufacture antibody and Macrophages are stimulated to attack the offending pathogenic antigen. If it is a CD8+ T cell (cytotoxic T cell) it will become an activated cytotoxic T cell and directly attack the `foreign' antigen.
Indirect allogenic recognition occurs when a recipients antigen presenting cell expresses recipient MHC bound to donor antigen. The donor antigen is usually catabolised donor MHC. This may come from processing of shed cells from the donor tissue or via the effects of recipient cells which invade the donor tissues. CD4+ T cells (T-helper) bind to the recipients antigen presenting cell expressing recipient MHC bound to the donor alloantigen. This releases cytokines to stimulate the immune response of the delayed hypersensitivity type.