If a minor faction (MF) is present in two or more systems and at War or Civil War in one of them, then the other (systems) that it's present in will see a reduction in INF if nothing is done BUT only if other MFs in those system move INF in a positive manner. I.e. if no other MF moves its INF then the INF of your chosen MF should not deteriorate.
To improve the INF of your chosen faction:
1. In the War/Civil War system, hand in combat bonds that will need (by definition) to have been earned in that system (or could theoretically be combat bonds retained from an earlier conflict involving the same MF). Can also submit routine bounty claims, earned in either the system in conflict, or the other systems it's present in (I presume this latter point to be the case, but not tested).
2. In the other system(s) its present in, hand in bounty claims. I can't recall (and haven't tested) whether combat bonds from the conflict system can be surrendered in a different system.
Bounty claims do not need to be earned in the system in which they are surrendered. E.g. if there's a Haz-RES in one of the systems this may generate bounties faster for your MF (with a KWS if not the controlling MF, or directly as the controlling MF with or without a KWS) and these bounty claims, once earned, can be surrendered in any other systems. This may be quicker that seaching for Wanteds at NAV Beacons or to interdict in SC.
Frustratingly, for both bounty claims or combat bonds their surrender is all or nothing. I.e. not possible to split up a larger pot of available bounty claims across two or more systems.