The reality of modern warfare doesn't favor large forces, poorly organized and underequipped, that are attacking reasonably well prepared defense positions.

Now, is there a will to use all the tools of modern warfare against climate refugees? Currently, no. But if "hordes of climate refugees will destroy the first world countries" stopped being a distant theoretical concern, and became a practical one? If there were real examples of border checkpoints in first world countries being breached by force, with border security overran, and thousands of somewhat armed and somewhat violent climate refugees pouring in through the breach? I expect that to change very quickly.