Outside Mexico, at places like Cosme and Empellón in New York; Hoja Santa, the Adrià brothers’ restaurant in Barcelona; Broken Spanish in Los Angeles; and Cala and Californios in San Francisco, chefs are carefully combining Mexican flavors with modern ideas and local references. At Atla, in Manhattan, the tostada with Arctic char, farmer’s cheese and capers deliberately echoes the Lower East Side’s traditional bagel with scallion cream cheese and lox.

The pop-up restaurant Noma Tulum, an extension of Noma in Copenhagen that is also known as Noma Mexico, is up and running here, built by the innovative chef René Redzepi, along with Ms. Sánchez, more than a hundred employees and dozens of local carpenters, metalworkers, farmers, ceramists and cooks. Until May 28, dishes like octopus sealed and steamed in fermented corn husks and a spicy-sweet dried pasilla Mixe chile stuffed with chocolate sorbet will be served under the stars by a cadre of earnest young people in linen uniforms and matching Birkenstocks.

Is this the “authentic” Mexican food that many admirers of the cuisine have tried to recreate? No. But the struggle over authenticity may be giving way to something more rewarding: a worldwide conversation about Mexican food that is respectful and inquisitive.

How did Mexican food, often viewed by those beyond the country’s borders as cheap, dull and heavy, move to being seen as artful, fresh and fascinating?

“It began with more Mexican people being able to travel, with the internet, with a younger generation who started to care about food being fresh and healthy,” said Gabriela Cámara, the chef and owner of several influential restaurants in Mexico City and Cala in San Francisco.

Ms. Cámara had no culinary training when she opened the restaurant Contramar in 1998 at age 23, to bring fresh seafood from the coasts to landlocked Mexico City. At that time, most ambitious restaurants in the capital were French or Italian. “Even Montezuma had fish from Veracruz carried to him in relays,” she said. “We only had frozen fillets from the Mediterranean.”