It really depends on the squadron and service.

My background USMC: it was on the flight orders signed by the CO the day prior. Cut and dried.

My background as a PIT instructor with the USAF: we were directed to determine it at the brief based on qualifications and time in the aircraft.

My background as a USAF U-2 pilot/T-38 pilot, we had specific rules put out by OGV delineating qualifications ( IP counted but EP didn't count) but not total time in aircraft. All things equal, rank was the determining factor.

The underlying sentiment (in both the USMC and USAF) was that the senior guy was going to be responsible for any unplanned or screwed up events. If we had a reason to make the senior guy the AC, we used it.