I imagine the champions that are flat out destructive (Brand, Kog'maw etc.) are creating isolated disasters, much like a pack of wolves making their own attacks in the wild. Brand seems to be rather intelligent and is aware that he can't just destroy everything since he's clearly capable of being overtaken as evidenced in his lore. Kog'maw is just hungry and wants to eat things, and likely acts much like Rek'sai in that he kills/eats out of necessity (or territory), not "just cuz." Although given his bottomless stomach, his "necessity" might resemble "just cuz."
The other champions who are more politically aligned (and even those who aren't), I figure they're biding their times, or making (or attempting to make) tactical strikes against their enemies, or are going about their own lives (Karma is a spiritual leader for instance, Gragas is enjoying a nice ale, Udyr is battling inner/outer demons, Braum is busy brushing his and his poros' mustaches etc.).
Other champions are doing their own secretive stuff. Elise is seducing and convincing various people to go and be gobbled up by a giant spider, Veigar is doing his own little evil plans here and there, and so on.
Frankly, the only area of Runeterra that should be in continuous war at the moment is The Freljord. The other city states and their champions can make momentary truces or cease fires for the sake of mutual stability, like Miss Fortune and Gangplank working together at the Battle of Knife Straits.
