It's the same family, but you are meeting them for the very first time in each scene. They represent the idealized American family of the respective era they are depicted in. They only age about 3-5 years in each scene, for variation.
Even in the 1964 world's fair version.....John was in his 30s in 1890 and by Act 4, the children were old enough to drive and pick up grandma and grandpa (who are still alive) at the city's new jet airport. Mother had hardly any wrinkles and may be experimenting with hair dye.
The family's last name is Peabody, and they live in Middleburg USA, though a rep from GE in 1963 asked if they could be from Schenectady, NY...the original headquarters of GE.