Luna was born in Toluca and raised in Mexico City. His mother, Fiona Alexander, was a British costume designer, who worked in the film industry and made sure that it was a life in which he would be immersed; Fiona was of Scottish and English descent. She died in a car accident when Luna was two years old. His father, Alejandro Luna, is an acclaimed theatre, film and opera set designer, who reinforced the importance of theatre and the arts in his life. Alejandro would bring Diego to the sets and mentor him in different aspects of art, cultivating a desire to become an actor and uphold the family tradition.