The process is as follows:
- if the faction issuing the bounty is in the system where the bounty is turned in, it gets an influence boost in that system (regardless of where the bounty was awarded)
- if the faction issuing the bounty is not in the system where the bounty is turned in, it gets no influence effect in any system (but you will gain rep with the faction)
Turning in at a carrier might give a reduced effect in the first case because the carrier's cut of the hand-in won't be counted, but I don't think this has been tested recently and may have changed since it was originally tested. It does still give at least some effect.
So to go through a worked example:
- you support faction A
- your home system contains factions A, B, C and D
- you go on a mission to another system and come back with bounties for factions B and E
- if you hand those bounties in anywhere in your system, then B will get an influence boost in your system (and E gets nothing because they're not here)
- nothing happens to the influences in the other system
- as B gets an influence boost, then that will reduce the influences of A, C and D in your system
(Aside: this doesn't apply to trade or exploration; those action types always affect the station owner, and as a carrier's station owner isn't an influence-holding faction, trade and exploration at a carrier always has no influence effects)
