On many instruments, the combination action features multiple "memory levels" (possibly hundreds!). Each memory level is an independent copy of the settings for all the user-programmable pistons, so e.g. the "general 1" piston can be programmed to recall one registration on memory level 1 and a different registration on memory level 2. There's usually an LED display or screen that shows the active memory level (they're named numerically), with accompanying buttons to increase and decrease it. Generally, on a shared instrument (e.g. at a music school or large church), an organist will be assigned a range of memory levels for their own use. Some combination actions even have a "lock" feature that makes a memory level read-only until it's explicitly unlocked, which helps prevent one organist from clobbering another's registrations if they accidentally punch in the wrong memory level.
Some older instruments (built before electronics) also have combination actions with varying designs and degrees of flexibility. I don't know much about them, but the Wikipedia link overviews some of them. Indeed, older systems often need to be set in the "guts of the organ." They might even require an organ builder to come configure them.
Some organs (e.g. small instruments, Baroque-era instruments, and modern instruments built in certain historical styles) don't have combination actions at all. In that case, all stops must be moved manually, either by the organist or by a helper ("registrant") or two. It can take time to get used to playing while a helper on either side of you manipulates the stops. It can also be nerve-wracking to be a registrant, knowing that if you lose your place in the music, you might mess up your colleague's performance!
[0] https://en.wikipedia.org/wiki/Registration_(organ)#Combinati...