The president-elect has not specified how he will convince other countries to take back their citizens, but there is a law he can use to deny visas to uncooperative nations
That's definitely the play. It's another power (and money-) grab by the super rich: https://www.thelondoneconomic.com/opinion/brexit-a-movement-for-the-super-rich-by-the-super-rich-159708/ So why not try the same thing a second time? Then it was mainly bankers, now its tech bros. In both cases they don't really care about destroying the economy of one country. If they can't squeeze any more money out of it, they just leave like the locusts they are. The US is a particularly juicy fruit for them, it might take them decades to fully turn it into Russia 2.0.
Also most undocumented people in America come over legally through the 90 day tourist window or work/school visas, and just overstay their alloted time. Trump would have to ban all entry into America to stop illegal immigrants. Just ask anyone if that's even politically, economically, or socially possible.
While many Americans don't leave the country, increasing the cost of doing business abroad is probably the play. A lot of companies have outsourced teams in the Phillipines or Argentina etc