I've started John Darwin, After Tamerlane: The Rise and Fall of Global Empires, 1400-2000. The broad outlines of Western imperialism are familiar to me, though his theses put quite a different interpretation on them.
The History of China Podcast by Chris Stewart (/u/cthulhushrugged), even just up to the fall of Tang (as far as I've gotten), had rather a lot about central Asian barbarians. The History of Byzantium podcast from Robin Pierson has touched on them. I realized that I know little about them.
A book covering empires up to 1405 would be massive and incoherent, but are there any, maybe one just covering the bigger ones? Or any good book on large central Asian polities when they went and broke all their neighbors' nice things, or Transoxiana: What the Hell Was It and What the Hell Was There?, or something?
There's a book, (which I actually didn't like at all, and which isn't quite what you're looking for) called Empires in World History by Jane Burbank and Frederik Cooper. They have a chapter on the Mongols, a couple on China, Russia, and a lot of other pre-Tamerlane Empires. Like I said, I don't love the book, and I can get into why if you'd like. BUT it definitely gives an overview of those empires, and each chapter comes with a 'sources' section that gives the reader lots of avenues to follow up on.