I believe the policy in both CX and KA is that a Management pilot has to be the Commander for a delivery flight.
The remainder of the cockpit crew can be anyone qualified. Each fleet office has a request list, and if you don't put your name on the list then you are unlikely to get a delivery flight. Having your name on the list does not give you "first come, first served" status, since crews seem to be a balance of long serving trainers/checkers and the "chosen few".
I have heard from some crews that a "delivery" is a nice break from the routine, but not just a simple "trip out with the lad's" (and/or ladies) to pick up a new jet. Many trips require dinners/cocktails etc with Management/Manufacturer/Invited guests etc. Any face time with Management, is a game of chance where you run the risk of getting your foot stuck in your mouth!
Cabin crew selected for the flight are the "chosen few"