as the war progressed, did American units in prior campaigns and with combat experience get moved around throughout the theater to spearhead offensives and invasions elsewhere?
The best answer to this is to pick a unit of your choice and look up its battle history. A lot of units, especially those established early on, saw service in multiple theaters of war.
The 1st division f.x. fought in Africa as part of Operation Torch, then went to Sicily and finally ended up in France.
Bear in mind that turn-over would be significant, the division posted about 20.000 casualties but a hard core of veteran men and officers would have remained to bolster the unit.