If people start migrating away from those regions en masse, is there a chance it will trigger a war?
Wars in Africa can be traced to shrinking rivers and lakes. Tensions between north and south Sudan, Uganda and Rwanda,
When you see pictures of people starving, they never hold AK-47 in their hand.
Historically this is a recipe for raising cheap armies, and finding something to do with them.
In another viewpoint on climate change, do remember that the Earth rotate around the Sun in elliptical manner, it's not perfect circle - there will be surely a time when a new Ice Age will happen - we're just at a period when Earth is getting closer to the Sun on it's rotational axis
The earth's orbit around the sun is an ellipse. But it's an ellipse that takes one year to traverse. The closest thing that gets us in terms of an "ice age" is winter. And, in fact, "winter" is what happens when you're closer to the sun, due to a change in the angle of incidence of sunlight (in the northern hemisphere).
Are you referring to something else?