In my experience, in all other systems that are not on war, missions, passengers and trade work very well.
Trade works also in the war system very well if traded reactive armor, personal weapons or battle weapons.
Also in the war system works very well war specific missions like surface scan or assassinating. We did sometimes no combat but only such sort of missioms and got a good boost in influence (traffic was watched with eagle eye so that random people coming around weren't disturbing the results).
Edit:
Evidence is following
Our PMF is currently at war in one system. We have chosen another systems where we do only missions and trade today, no bountys or combat bonds (combat bonds have outside of the war system anyways no impact on influence, only to rep).
I will edit this post and publish our data to proof. Expect it to around mid-day tomorrow after the BGS tick.
Edit2
It will take one tick more sadly. My Hotas has broken in the evening and have to got to the store to get a new one. Won't be back before tick

Good news: one day of war is remaining. Our PMF has lost influence and gained influence i other systems than the war system. So i was already able to disproof statement about frozen influences in the other systems.
Lots of changes to be seen. Posts will follow as soon as i have collected everything
Time until new Hotas: 2h