Would a blood transfusion affect immunity?

Yes. Transfusion-associated immuno-modulation (TRIM) is independent of whether the blood is allogenic, autogenic, or leukodepleted. This has been described since 1973, seen in the improved graft survival in kidney transplant in transfused patients. Some effects include down-regulation of the immune function by decreasing cytokine production, and an increase in T suppressor cell number and function.