NEW DELHI — India has a teeming population of able-bodied workers, tens of millions more than its employers can accommodate. Many other countries have the opposite problem: more jobs than workers. Today, across India’s government and business sectors, a movement is gaining steam to begin exporting more workers. The idea, which economists call labor mobility, is to connect young Indians to companies in places with shrinking populations where labor shortages are holding back growth. The challenge for India and its partners overseas is the growing opposition to immigration in many countries. Officials are trying to craft policies to make it...