With border tensions between India and China at their worst in decades, Indian Defense Minister Rajnath Singh headed late last month to a familiar destination: Moscow. There Mr. Singh finalized a $2.4 billion deal for 33 fighter aircraft, and pressed the Russians to speed up delivery of the S-400 missile air defense system, which India bought two years ago despite the threat of U.S. sanctions.

The defense minister also watched a contingent of Indian troops—among a handful representing countries not formerly part of the Soviet…