Trying to force a minor faction into a retreat, they are in elections in a neighboring system. Will destroying their ships in the system I'm trying to kick them out from lower their influence? I know missions have no effect, but murder?
Missions will have an effect during an election. Supposedly, combat actions do not. You wouldn't be ale to help them with bountyhunting, but you would be able to help them with missions. Though this is little help to you trying to force them down.
One thing that does help, is that bounties for the same superpower won't help a faction in election, but does help any other factions of the same superpower.