As Russians escaping a partial military mobilization in their homeland drank their morning coffee in a cinema that opened its doors for them in northwestern Kazakhstan, Central Asia was waking up to yet more fallout from Moscow’s bloody war in Ukraine. Almost a week into the drive for soldiers that Russian President Vladimir Putin announced on September 21 in response to a tide-shifting Ukrainian counteroffensive, sightings of Russians wandering the streets of cities throughout the region with backpacks and lost expressions have become commonplace. So, too, have lengthening lines at the state borders that Russian citizens can still cross without...