Speaking of historical accuracy, in age of pirates you can enslave enemy crew members, however I am not quite sure it was the case. While caribbean-european born or descendant (or any other ethnicity for that matter)- christian-pirates could and probably did steal non-christian black-moor-slaves from slave merchants, I am not quite convinced such a pirates would enslave free Christians, whether white, black or else, because they would have great difficulty selling them over.
Most people forget to take into account, or just don't know at all, that religion was an important factor (the factor, really) in XVth to XVIIIth centuries European slave trade. Racist ideas started to develop only in the XVIth century, until then there was just ethnocentrism, race becoming the predominant factor in western slavery just in the XIXth century, maybe beginning in the late XVIIIth. Then again, in the XIXth, when race became the main factor, christian missionary activity towards black slaves increased and the abolitionist movement (mostly Christian-driven, but with humanist influence to some extent as well) grew to the point of, once again, abolishing the institution of slavery in the Western World. Muslim pirates from the Barbary Coast and the Ottoman navy, however, did raid European ships and Mediterranean ports taking Christians as slaves by force regardless of ethnicity. As far as I know, Europeans bought most, if not all, of their slaves (already enslaved people) from Muslims and Pagan Africans. If by case it was a christian slave being bought they'd probably free him or turn him into a serf.
Christians since the roman empire had the tradition of freeing up christian slaves they might have own, as such, with Europe's conversion to Christianity, slavery was effectively abolished from Europe, developing in it's place feudal serfdom. The serf not being considered a commodity (like a slave) was attached to the land and had a series of rights which slaves didn't.
In the modern era, in the growing cities and in the America's, a "new" kind of servitude emerged; indentured servitude. Young people that had no marketable skills and or people that wanted to settle in the new world, but didn't have the money for paying the voyage, or the skills to become a sailor, could sign a contract to become a serf for a master that could teach them a profession or take them to the America's in their ships, in exchange for their labor for a certain period. In this kind of servitude, the master could sell the contract's rights to another person, which was what captains did as soon as they arrived the New World, with the selling of contracts paying up the costs of the voyage and giving some profit and the new master being the actual beneficiary of the serf's workforce.
This encouraged good conditions in the ships transporting this kind of people, maybe due to contract clauses and people not wanting to buy the rights to sick-ill treated serfs. Later, when indentured servitude went into decline and was abolished, the voyage conditions in the ships transporting people that couldn't pay declined sharply.