The vast majority of slaves transported across the Atlantic were not enslaved by Europeans. European slavers did not descend on Africa and kidnap free Africans, they purchased them from African governments who had already enslaved them. One of the reasons that so many slaves came from what is now the Niger delta is that there were city states there with large slave markets.

I don't want to sound like I am downplaying Europeans' role here. Europeans didn't simply act as transporters in a pre-existing system of slavery. There is ample evidence that, when they saw the numbers of slaves that Europeans wanted to buy and the prices they were willing to pay, African rulers took steps to greatly increase the number of people they were enslaving. There are cases where wars were started specifically in order to get slaves to sell to Europeans, and during the years of the Triangle Trade, many other communities increased the number of crimes for which slavery was available as a punishment.

Non-Europeans were also involved in the transport of slaves across the Atlantic as crew members on slave ships.

Source: Edward Alpers, 'Ivory and Slaves in East Central Africa'.