Right now, many of the motions are subtle, because people like where they live and often creep rather than run away from danger. However, as climate change worsens, we will see more and more areas become uninhabitable due to heat, drought, flooding, etc. Those people will ALL have to move.
Furthermore, big disasters can drive big movements as people flee the scene en masse. Over the next few decades, we'll see a lot of that, and it will often be city-sized groups in the thousands or millions. They'll stick with their own country if they can, but the smaller the country, the less running room it has. Island nations are screwed, sure, but so are many small land nations. They just don't have anywhere to retreat, their whole territory is ground zero. So people flee to other nations. Forget defending the borders against truly large migrations of people who have nowhere to go back to.
What few people are paying any attention to is the fact that the hot parts of the planet will become uninhabitable. Because climate news is routinely bad -- "We underestimated this effect and it's much worse than we thought" -- that's likely to happen a lot sooner than a century from now. 1-2 billion people fleeing environmental foreclosure is probably an underestimate.
Of course, you also have to account for escapability. Some things like a heat wave can get so bad, so fast, that many people can't get out of range before they die. Tidal waves, earthquakes, those can do the same. Hurricanes usually give a few days' warning, but then you have to consider whether people have anywhere else to go and roads to get there. You're always going to lose some people, and sometimes, a lot of people. But that's harder to estimate than the number of people under threat, which makes it almost impossible to extract meaningful numbers of surviving refugees -- which is the number people need to know in order to prepare.
My best advice is just expect that people will want to leave the middle of the planet as it cooks off, along with drawing back from low coasts, and try to arrange routes and refuges accordingly.