Kestrel Update
It's conclusive: The birds we saw are Common Kestrels. A photo, attached here, of the black claws removes any vestiges of doubt. (I'd ignored that photo until earlier today.)
I am unfamiliar with Lesser Kestrels, having seen them, most unforgettably, only in Ronda, Spain in 2004. That's why it took a while for me to be sure. When I happened upon the shot attached here of the black claws, finally I had conclusive evidence.
Here are some of the other factors that were pointing toward Common and not Lesser:
1. Spots: None of the books I consulted mentions any form of male Lesser Kestrel EVER having ANY spots on the mantle, back, and lesser coverts. Our bird obviously has spots on the mantle, back, and lesser coverts.
2. Breeding habits: None of the books emphasizes major differences in breeding locations between Lesser and Common. The biggest difference, running through all the works I consulted, is that Lesser tends to breed colonially ("normally in groups of no more than 25 pairs"--HBW II), but Commons, though they may breed colonially, are less likely to do so. A Lesser Kestrel is likely to breed on a cliff face (as was the case with our kestrels), but so is a Common.
3. ALL books I consulted mention the black claws of Common and the white or pale claws of Lesser. The closeup takeoff shot clearly shows black claws.
4. Range: HBW II on Lesser Kestrel: "In Asia regularly breeds up to 1500 m." HBW II on Common: "[Breeds at altitudes] occasionally reaching 4500 m." Our birds were found at 2600 m, well above the limit proposed for Lesser in HBW II and well within the limit for Common.
5. Whether through maps only or in the written description, MacKinnon, Raptors of the World, and HBW II all have Lesser Kestrel breeding north of Gansu. MacKinnon: "Breeds in N and W Xinjiang, Nei Mongol and Hebei." Common Kestrel breeds over much of China, "except dry deserts." Our area was indeed a desert or near desert, but a stream was nearby; most likely MacKinnon is talking about the really dry deserts like Taklimakan.
Works Consulted
Collins Bird Guide to Britain and Europe
HBW II
Raptors of the World (Ferguson-Lees, Christie)
MacKinnon (A Field Guide to the Birds of China)
Birds of the Indian Subcontinent (Grimmett, Inskipp, Inskipp)