First, weren't those past changes much slower, thus giving much more time to adapt?
Second, wasn't the world much more sparsely populated, meaning that if your area's ecosystem changed it was often feasible to migrate to a most hospitable currently unoccupied area?
Third, if they did migrate to someplace that was already occupied and those already there objected, nobody had weapons of mass destruction. At worst, one group would genocide the other, but it wouldn't spread beyond the disputants.
If a climate change induced resource problem leads to war in our times there are weapons of mass destruction, and those wars could be very devastating and draw in others.