Does a Republican automatically turn into a Democrat or Independent the instant they announce the switch, or are there forms they need to fill out? 
Aside from the elbow rubbing, they probably do what you and I do - reregister as the other party. Though I dont know if this means you are a full-fledged member and will get the newsletter and learn the secret handshake etc.
It’s not much more involved than saying “Hey, I want to switch teams.”
It’s not like the membership qualifications are strict for either party.