Matt Pyken knows how to tell a story. Over a 20-year career as a writer, he has worked on broadcast TV dramas including “Empire” and “Castle,” and consulted on other shows, including “Mr. Robot.” But over the past few months he had an entirely new assignment: helping TV writers in India learn to write for a global audience.

Netflix hired Mr. Pyken to work as a consultant on its upcoming drama “Leila,” currently being filmed in India, about a mother searching for her missing daughter. He is one of several American producers Netflix pays to work with writers on shows it is making in different countries, including Thailand and Korea. Its aim is to bring more of the U.S.-style of serialized drama—where episodes of each season follow an overall storyline (think “Stranger Things” or “Game of Thrones”)—to shows made for foreign audiences.