The office crossed the Atlantic with the people who held it and kept its structure while changing its vocabulary. In Brazilian Candomble the babalorisha is the babalorixa, more commonly called the pai de santo, father of saint, in the Portuguese that the tradition's public language became. In Cuban Regla de Ocha he is the santero, and where he stands as initiator to a particular person he is that person's padrino, godfather. The Cuban houses also distinguish the oriate, the specialist who directs the ceremonies, which is a role Yorubaland does not separate out in the same way.
The lineage logic survived all of it. A Havana or Salvador house can still be traced by naming who initiated whom back through the generations, and legitimacy disputes in the diaspora are almost always disputes about that chain: whether a claimed initiation happened, whether the initiator held what he claimed to hold, whether a line has been broken.
Since the middle of the twentieth century there has also been traffic in the other direction, with American and Brazilian priests seeking initiation or reinitiation in Nigeria in order to attach themselves to an African line. That has produced real friction, because a Nigerian house and a Cuban house do not necessarily accept each other's reckoning, and the question of who is authorised to say so has no single answer.