All you do then is turn the entire country into an "attractor country" - faced with the 'homeless problem' of the rest of the world. Which is, ironically, one of the issues raised against open borders.
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.
X% of the population will always need some gov't help and Y% will need gov't help at some point. I think we need to stop negatively viewing these groups as sponges on society. Ability is a spectrum, and we don't criticize someone w Down's Syndrome, and so I don't think we should be so harsh on people who are homeless / have trouble holding down a job / low-ish IQ / and even those who fried their brain w drugs / etc.
Yes, we want to avoid perverse incentives and incentivize people to be productive members of society, but we also need to recognize that some people will always need some help, and as a society, I think we should help them, striving for a future with little to no involuntary poverty.