In cases of systemic violence like racially-motivated lynchings, I don't believe it is the lack of empathy that's at the root of the problem, it's the dehumanization of entire classes of people and the backlash against those who oppose the rules and practices that protect civilization -- as those in power define it.
Look at the laws (now all gone, hopefully) prohibiting marriages between a person considered white and a person of any other race. The traditional definition is that marriage is a covenant between one man and one woman. So if one of the people wanting to be married isn't considered a full person how could marriage between them be any more legal than a marriage between a man and a mule?
Once we stop viewing people as fully human, we can justify all sorts of actions that would otherwise be sickening. Before empathy can exist we need to see those we lack empathy for as full human beings no different fundamentally from ourselves.
If we can't do this, the problem lies within ourselves.
Rachel's post on Alas, a blog