I’m going to guess a fleet carrier will be a similar to The Gnosis. It will move once per week at the tick - like other mega ships. The Squadron Commander (and selected officers) will select the new location.
It's a conservative but logical supposition.
I hope it can jump in Witchspace anytime in a similar fashion to the Capital Ships, making jumps up to 500 LY (like the megaships) and that it drops directly in the orbit of the destination body (not at the main star). To balance the game it then would require a long cool down: 12 to 24 hours, depending on the jump distance.
To add gameplay: it could be refuelled with gas scooped by the regular ships from the gas giants. This would require some big ships (like a Type 7's) to make it faster, equipped with dedicated fuel tanks and a dedicated fuel scoop. Of course the modules can be stored in the Fleet Carriers when the tanks are full.