My experience was when KUL moved from Subang (now SZB) to the new KL International Airport (KLIA).
The moving date was declared way in advance; I seem to remember the date was locked more than six months before, and a lot of airlines rescheduled their operations, especially the night-stopping aircraft. Obviously, Malaysia Airlines had to ferry a bunch of planes over, but it was a 10-minute hop and done in the early hours, so it was not much of an issue. Some of the ground equipment was ferried over earlier in the day (of the last day of Subang operations), but everything else was moved over once the last flight of the day was completed. I'm talking motorized stairs, K-loaders, belt-loaders, tractors, trolleys, dollies - the works. It was quite a convoy of flat loaders. Stuff that could be driven on public roads were given temporary permits, so you saw motorized steps and water/toilet trucks on the public highways!
The biggest change was moving from a host (MH) check-in environment to a homegrown common-use system which was integrated with the Baggage Handling System (BHS). The first day's baggage handling was chaos, with a lot of bags not making their flights. A lot of items can't be duplicated, not only in terms of equipment but also in manpower, and it is easier to make a clean cut and manage the problems for a 24-hour period than drawing the pain out over a period of weeks.