Societal collapse in for example India or Brazil would make things awkward real fast for a large chunk of the world
The military does not even play a role here. All the poorer countries need to do is a) not prevent their own young people from leaving, b) not prevent activity of smuggler gangs, c) complicate deportations by refusing to issue passports to their citizens who destroyed their papers underway. None of those actions need guns, much less superior guns.
[0] ... India and Brazil aren't really in the "top population growth" league. Look up population pyramids of Niger, Ethiopia, Nigeria, Yemen. Those graphs promise future mass migration streams. (https://www.populationpyramid.net)
[1] ... https://ec.europa.eu/eurostat/statistics-explained/index.php...
Your parent was suggesting that the developed world could not deal with a collapse of India or Brazil or several populous African countries.
What would happen if 20% of India decided that their society has collapsed and they should migrate at any cost to some developed country?
If traditional food sources go away and technology provides an alternative food source, that is what people will eat. Culture will follow technology, if that's what people have to do to survive.
EDIT:That mostly applies to the population in the developed world with access to capital, technology and industrial resources that we can use to adapt our food supply. Not everywhere in the world has those resources to fall back on though.