Thinking of those countries in South Asia and the Middle East that will likely be getting regular lethal wet bulb temperatures, or those coastal countries that will likely lose land to sea level rises

This will resolve to some extent on consensus, but such a war would be characterized by the seizing, or failure to seize, of cooler or higher territory.

Presumably before (and after) would be characterized by wet bulb events, or coastal flooding. It seems fair to expect that there would also be a public media debate about the need for such action, at least in countries with a free press.