Advocates who say black Americans should be compensated for slavery and its Jim Crow aftermath are quietly chalking up victories and gaining momentum. Fueled by the work of scholars and lawyers, their campaign has morphed in recent years from a fringe-group rallying cry into sophisticated, mainstream movement. Most recently, a pair of churches apologized for their part in the slave trade, and one is studying ways to repay black church members. The overall issue is hardly settled, even among black Americans: Some say that focusing on slavery shouldn't be a top priority or that it doesn't make sense to compensate...