Related to this - does supporting and raising the influence of the dominant faction in the system have any effect? My understanding was that the intent behind the background sim was that factions could eventually expand into neighbouring systems. But at the moment, this is borked?
This is working, though I don't know the exact mechanic for triggering expansion.
Whether this is a mechanic that will stay or not, in the PP Beta, all the factions are prefaced with their originating system name, so, the Sitakapa Empire League is now distinct from the, say, Facece Empire League.
I can verify the expansion mechanic (and subsequent invasion) works too. I flipped a system to the control of a faction which expanded into it (in this case, flipped control of Mac Og for the Sitakapa Empire League), and again, under the new Beta, it lists the controlling faction of Mac Og as "Sitakapa Empire League".
As others have stated though, get your supported faction into 70% influence, this guarantees triggering a war with the faction who owns the "system controlling" station. When you win the war (go fight in the conflict zones for your faction), you gain control of the "most valuable" station; Stations which determine system control are always considered most valuable, so you'll get that, and control of the system.
Just note, the galaxy map won't update, but the system map will. I don't know why, but this needs manual changing. Mac Og, despite being controlled by an Imperial Patronage government (there are now police and criminals are marked as wanted system-wide), the system is listed as an Independent Anarchy still.
EDIT: Also, the best time to run missions is when your faction has just expanded into a system and still has the "expansion" state. I was conferring a 9% influence change per day on my own when flipping Mac Og.