Nothing “simply” about it.
One million people is a single city. You do have a problem if one million people are suddenly relocating. It has happened before for war, natural disasters or other reasons.
Migrating a billion people is not a thing, for the simple fact that there isn't any human settlement any close to approaching a billion people. If you're talking about the scale of a whole continent, migratory flux span over a lifetime and are not homogeneous.
Yet.
I suspect you may be right that the popular option will be to close the border. I do not agree however, that it is the better option.
Interesting way to describe people fleeing war.
A crop failure might be a natural disaster, but lots of other causes of starvation that are not.
Main point is, if you’re seeking refuge, try to put a clear-cut reason down rather than something debatable, but sensible, like absolute poverty.
This will certainly happen in a scenario where there are massive refugee populations elsewhere in the world.