I think in many cases, they stay for the military and all the others who've supported the regime for years.
The dictators may be able to exile themselves and their families, assuming they've not committed war crimes or are wanted by the ICC and assuming some country is willing to host them.
But they might not be able to leave behind those who made their power possible. Supposedly that was a factor in Egypt, as a reason Mubarak didn't leave. I think the US would have facilitated his exile just to avoid bloodshed on the streets.