I'm reasonably well off, childless, and if I were given a free choice of any country to emigrate to, I'm fairly sure the U.S. wouldn't be the first on my list.
There's a case to be made that welfare states are immoral for this reason. If national social guarantees force you to restrict third-world economic mobility, it's not progressive: it's the opposite. Internally it's egalitarian, but globally, it's a group of rich people using force to preserve inequality. It's not just failing to help the poor: it's actively repressing their attempts to help themselves.
I've always thought of this in relation to the outcry against "the 1%" in the US from people outside the country (like Scandinavia, etc). We might be unequal in the US but the One-percenters don't have the right to use deadly force and do not have standing armies. But these so-called egalitarian northern European states have both. (Although I do in fact love Northern European countries and their people, I'm just pointing out the strangeness of the position.)
In many ways we'd see a massive US if borders in all countries were opened. The author of the piece hints at this as well. Many groups will still lose out, but the laws will be uniform and will add to efficiency. I feel like the British Colonies were a good example of this.
I believe most of the people, especially the most poverty stricken ones, would stay put
How would they move if they can't support themselves? Food is the priority
Of course, this may lead to a different balance of people in countries, so it may work from that side.
Well as a citizen of two countries (one in the EU) I am quite mobile myself, but sometimes the language is a bigger barrier than the 'official' borders.
On the other hand, some people go out of their way to cross borders illegally exactly because it's illegal.