The Civopedia doesn't provide much information on how "war weariness" relates to location except to say that it is applied to every city you own, and that it is greater when fighting occurs in foreign territory.
There seems to be something more to it though. I've been defending successive invasions from a couple of other civs, and 4 of my cities have 0 war weariness, and one has -6 war weariness. The city around which most fighting occurred has 0 war weariness, the next closest city has -6 war weariness, the two others on the same continent have 0, and a city on another continent has 0.
While I say I've been "defending," much of the fighting happened within range of a nearby city owned by the opposing civ (the cities are only a few spaces apart), so some of this could be considered foreign fighting.
This looks more like a fluke than a game mechanic because there's nothing special about the city with -6, but I could be wrong.