I would imagine that a catastrophic and sustained rise in temperatures would disrupt agriculture, reducing food production to the point where people starve to d

I would imagine that a catastrophic and sustained rise in temperatures would disrupt agriculture, reducing food production to the point where people starve to death before they roast to death. And in practice people don't just sit there and starve to death - they migrate, with the cooperation of other nations or not. A lot depends on the rate at which migrants arrive, and where they go. If the migration is towards northern Asia, spread out over 100 years, with only a few tens of thousands arriving each day, Russia will probably be able to manufacture bullets in sufficient quantity. If 1.5 billion Indians descend all at once, nothing will be able to stop them. These are extreme outcomes. There are so many other possibilities that could be negotiated. Perhaps India will gain economic leverage through exporting solar power, and be able to strike a deal to resettle some of its migrant population elsewhere.