Students fill out an LOA (leave of absence). FSA takes your name out of the computer for the days you put down. Nothing to it. Its the same thing for leaving on long weekends and thanksgiving. If a student needs to go home for a month, not a problem (usually), just fill out an LOA.
Instructors can also go on LOA. Normally when you go on LOA you just fill out a form listing all your students and who is going to be taking care of them. If your LOA is just for a few days, you still need to list instructors to take care of your students, but if your students don't mind taking a day or two off, they don't have to fly.
For Christmas, you have the choice of 2 standardized LOAs. You can leave a little earlier and be back before the 1st of the year or leave a little later and have New Years off. In reality it doesn't really matter, since almost all of the students leave for 2-3 weeks. This is the only time you don't have to deal with finding someone to take care of your students. If a student wants to fly over the holidays, an instructor who is staying in town is assigned to them.