Homeless people are generally not allowed to cross borders pretty much anywhere in the world, except under special conditions (e.g. being considered a refugee fleeing a warzone or failed state.)

This is why most countries' border services will ask short-term visitors the address of the place they'll be staying while they're there: no country, no matter how open its borders, wants you to visit with the goal of becoming a beggar on their streets.

(And this is also a large part of why particular passports only let you visit particular places without a visa. The EU has internally-open borders for EU citizens, because they have mostly equalized levels of social-welfare "pressure" between countries. But no EU country will let you in from a non-EU developing country that has high levels of homelessness, if you can't prove income; and that's something the EU enforces on its member states, so that no country becomes the de-facto homeless ingress point for the whole of the EU.)

Because of these measures, each country (or open-bordered region of equal-welfare countries) is mostly just dealing with its own homeless problem, not with other countries'/regions' homeless problems.