I think it is safe to make the assumption (yes, it makes an ass out of me and umption) that trans-generation slavery does not exist anymore, as it did in ancient Rome or the antebellum South.
Since slavery is universally outlawed, it must exist in secret. A governmental body would have to recognize legal ownership of human beings in order for multiple generations to exist in slavery. Such an institution isn’t practical in the industrialized world, to say nothing of the human rights issues involved. No nation could get away with it without being ostracized by the rest of the world.
That is why overt slavery today is ad hoc and temporary. You can get away with it for a while, but eventually you might be discovered. Child slave labor is popular because children are easy to control and exploit; but they eventually grow up. POW’s have been used as slave labor, but usually only for the duration of conflict.
Nations have found that you can get away with slavery longer if you can gloss it over. If you can force a person to work against his will, but pay a token wage, then ugly words like “slavery” can be avoided, and pacifying terms can be substituted like “national conscription,” “dictatorship of the proletariat,” or “income taxes.”