It really depends, if they get lucky they might end up being raised in a country that has an easy path to citizenship for people who pass through the education system and learn the local language (for more niche languages this is especially true).

Anyways, there is a significant number of people who end up stateless and life is really fucking hard. You have no passport and to cross any border you need to manually apply for a visa with absolutely no guarantee of being able to return. Imagine if you went to Cancun for spring break and the US just said “No” when you tried to go home - all your shit is (pretty much) lost to you except what you have on your back and Mexico will eventually kick you out into a country with weaker border control. It absolutely fucking sucks and, while I appreciate that my life has been made significantly easier by being born in the US I think it’s unethical that birth location has such a strong effect on our lives.

So, usually, those folks just become stateless and get lost in the cracks of bureaucracy. If you’re not a citizen, and especially if you have an illegal status (that would be likely for such an orphan) you won’t have access to healthcare or social services, you’d risk deportation if you’re ever (even wrongfully) arrested. You’re playing the world’s most depressing game of roulette until you get noticed and lose everything.

It’s for reasons like this that the anti-immigration crowd really fucking pisses me off.