Where do illicit artifacts usually end up? Is there a type of person who buys these? Drug cartels? Russian oligarchs?

The British Museum has historically been a common destination. Other Western museums aren't far behind - although it has become less common over the last few decades.

Illicit artifacts usually end up in first world countries in surprisingly normal homes. It's pretty normalized. You may have come across things in antiques stores, or art galleries or coins on eBay.

Small scale looters steal the artifacts, then they're laundered across borders by organized crime as a funding source, and they don't care much where things end up.