There is virtually none there to begin with. Only the white blood cells have a nucleus, so they are the only that carry any of the donor’s DNA. Red blood cells and platelets lose their nucleus during production in the bone marrow.
Donated blood is spun in a centrifuge to separate it into plasma, platelets, red cells and white cells and only the first three are used for transfusions. If whole blood is used in an emergency transfusion, it causes a fever called ‘febrile non-haemolytic transfusion reaction’, as the recipient’s own white cells destroy the foreign DNA.