Which countries attempt to attract their diasporic or emigrant communities to move there? How can they do that? (living)

[markovian process]

I recall reading about how some countries, like Italy or Poland, offer those with the same ethnic background that have ancestors/families who left the country and formed diasporas elsewhere, the chance to immigrate "back" to their country based on some ideas about kinship. Is this true? One example I've heard is Japan trying to get Japanese-Brazilians to immigrate "back to Japan" but the Japanese-Brazilians leaving after staying for a while, because of course they don't relate to the culture.

How does this kind of process work, for those countries that want to get their diaspora communities to "return"? If the immigration event was a century or two ago, are there that many people that have documentation to prove that their ancestors came from their "homeland"? I guess it works if you have someone within living memory like a parent or grandparent but further than that, how can you tell kinship/ethnic ties without documentation (it's not like you can just look at a person, right?).