‘They’re getting divorced, and they’ll do anything NOT to get custody of the kids.” So reads the promotional poster, in French, for a new movie, “Papa ou Maman” (“Daddy or Mommy”), plastered all over Paris during my recent visit there. The movie sounds like quintessential French comedy, but its plot touches on a deep and serious reality—and one not particular to France.

All around the world today, pre-existing family patterns are being upended by a revolutionary new force: the seemingly unstoppable quest for convenience by...