My tidbit to add is that when the Mongols got to Europe and encountered stone castles that were harder to siege, they solved the problem by bringing in seige engineers....from China.

They were doing that stuff on the reg. One of their demanded tributes was, essentially, skilled workers as slaves. They were very aware that they had a very particular skillset that didn't translate well to everything. So they took what they needed to run their empire and put it wherever it was needed - even if it was someone who knew how to e.g. do algebra.