Most U.S. foreign aid is credit to be spent with U.S. defense manufacturers. In other words, "foreign aid" is a way to hand out U.S. hardware to allies or those for whom it's an effective bribe. The best example of this is Egypt, which has enjoyed ~$3B annually in U.S. gear as the payoff for making and keeping peace with Israel since the Yom Kippur war in 1973.

So the money does stay in the U.S., at least.