Doubtless ‘extension’ or ‘generalization’, ‘development’ or ‘evolution’ fit the process you describe, and I suggest ‘campaign’ isn’t an example.
https://www.collinsdictionary.com/dictionary/french-english/campagne for one, supports my belief that ‘campagne’ no more means ‘battle ‘ in French than ‘campaign’ does in English. In either language a campaign is the series of battles that make up a war, and could mean ‘(a) battle’ only if by co-incidence the whole issue happened to be decided in a single fight. A campaign doesn’t have to include even one battle. It might be rare but campaigns and wars are best won by manoeuvre, not by blood-letting, which is why to go ‘on campaign’ means to take to the fields. It includes the readiness to accept a fight but it’s in the field, not the fight that we find the essence of the thing.
Almost identical etymology calls people who take their holidays under canvass ‘campers’ and explains both why demonstrators might ‘camp on your lawn’ and why that’s not quite the same as ‘parking their tanks on your lawn.’
The ‘countryside’ meaning might be lost in some dialects; it certainly isn’t in broad English.
Where I live in rural Suffolk, sugar beet farmers frequently say ‘campaign’ where other folk would use ‘harvest’, rather clearly meaning ‘to go to the fields and do all those beety, farmy, harvesty things that we do there at this time of year.’
I suggest - with no specific research - that the real derivation of commercial or political ‘campaign’ is not to battle for sales or votes, but simply to go out into the field…