Just rechecked our records for one of our factions during a recent WAR state. This faction is in 19 systems checked daily, and the influence rose in 3 of them.
Another one of our factions was in ELECTION and we were able to boost influence in another system.
It may be that combat bonds only have influence effect in the system that they are earned in. A bug report might be in order
Be interesting to see someone else do the same, seeing as you have seen a rise in 3 systems. Is that 19 systems with no other traffic to influence them?