Place kickers wear a two sizes too small shoe on their kicking foot. They also remove the spikes so as not to get hung up in the turf as they sweep their foot to the ball. I saw this during an interview with a place kicker. Punters I am not sure about.
Don't know about NFL but I do know that rugby players don't change their shoes (there is no place kicking in Australian Rules football). Punts, drop kicks, place kicks are all commonplace in rugby. Same guy(s), same shoes. And everybody plays the full 80 minutes unless injured, they don't replace the whole team every time possession changes.
Element 99 