Usually the punter will simply take the snap and look to throw, but more rarely the punter may run, or the ball may be snapped to someone else (usually a halfback or fullback) who then runs down field or throws.
Usually the punter will simply take the snap and look to throw a pass or run with the ball after the defenders have turned downfield to block for the punt return. In another variation, the ball may be snapped directly to one of the up-backs (usually a halfback or fullback) who then runs downfield or throws.