She hid with her four boys under the cover of darkness, the distant shouts of Border Patrol agents and a helicopter droning over the hills on the Tijuana-San Diego border. The agents threatened to end a one-way trip from Mexico to the United States with arrest and deportation.

Some 37 years later, Leonor Dávila, a U.S. citizen, says she’s grateful for the life she has built for her family in Chicago. It’s a far cry from the desolate ranches that raised her in the Mexican state of Zacatecas, where opportunities were as sparse as the weathered adobe structures that dotted the countryside.

So it came as a surprise three years ago when her daughter, Jenny Aguayo-Frausto, who was born in the United States, told her that she and her husband were packing up their lives to pursue a future in Mexico. For Ms. Dávila, the move was as perplexing as it was ironic.

“There are so many people who would want to come over here to the United States, and then there’s them, who don’t want to be here anymore,” Ms. Dávila, 64, said in Spanish.

Ms. Aguayo-Frausto, 30, and her husband, Kevin Frausto, 36, are part of a contingent of Americans of Mexican descent who, because of their ancestry, are becoming citizens of both countries — formalizing their Mexican American identity some 26 years after Mexico began recognizing dual citizens.

Some simply want the bragging rights of having two passports, which can also open up travel opportunities to countries with more troubled relations with the United States. Many others, like the Fraustos, however, see Mexico beckoning as a viable alternative to life in the United States.