How does South Park animate celebrity faces?

I know this seems like a strange question, but I am very baffled by how Matt Stone and Trey Parker managed to pull off those mouth and facial animations of Mel Gibson in "Passion of the Jew." In the South Park movie, the animations were there but they were not nearly as refined for Saddam Hussein, and most of the times they just resorted to the flip-top animation used for the Canadian caricatures in the show, as seen in Terrance and Phillips as well as Kyle's little brother. It doesn't look like the cargo clutch for Robert Smigel's segments in Conan O'Brien's show either, so how did they pull off this technical wizardry?

I think they just get a picture of the person and cut it so it has a mouth and just move it up and down making it look like it talks well thats what i would assume they do for some of them im not sure so plz dont flame me

I know the first few episodes they were just construction paper cut outs. I think that has been computerized now.

Same technique but they do use completely digital methods now.