Baron von Beer
Each airfield is attached to a specific unit (Fliegerkorps, JAFU, etc). When you rebase a unit, they will then fall under whatever unit that base is attached to. Eg: Move a unit from Luftflotte 3's Fliegerkorps IV, to a base attached to Fliegerkorps I, Luftflotte 2, then the unit will now fall under the latters command.
Correct about escorts, only work with their own Luftflotte. Likewise, bombers can only fly with units from their current Fliegerkorps (Or JAFU if you transfer them to such a base). The one exception is your KGr.100 units. They can fly lead for any bombers (possibly only night bombing, that's what they're for, and all I've used them for, never tried if it carries to daylight raids too). However, if you move them to another base, they lose this ability and can only lead from the unit their new base is attached. Moving them back to the original base will not change that, so don't move them unless you "mean it".
Operations points: Transferring costs 1 op point per aircraft in the unit. 40 planes in a full Geschwader, 40 points to move. So long as you have 1 point though, you can still move an entire unit. Not sure if this has any penalties, such as increased transfer time.