There aren’t “franchises” in most art forms. One exception might be architecture, where someone like Eero Saarinen worked under his father (Eliel) before eventually going out on his own. Frank Lloyd Wright had two sons, a granddaughter and a great-granddaughter who became architects.
Noel Blanc (sometimes called Mel Blanc Jr.) worked in as a voice actor for a while, following in the footsteps of his father. And of their have been a variety of generational acting families. The children of professional wrestlers often enter the business. The recently departed Randy ‘Macho Man’ Savage was a second generation wrestler. Randy Orton, currently a WWE champion is a 3rd generation wrestler. Not all family relationships among wrestlers are genuine though.
Washington Roebling followed in his father, John Roebling, and continued to build the Brooklyn Bridge.
The revival wasn’t all that bad, but it was nowhere near the greatness of the original.
The sons of professional base players have become stars themselves. Barry Bonds and Ken Griffey, Jr. come to mind. There have even been third-generation major leaguers, such as Aaron and Bret Boone, the sons of Bob Boone and grandsons of Ray Boone. Although, most of the time the fathers are alive to see their sons play, and in the case of Ken Griffey Senior and Junior, even became teammates.
Adrian Conan Doyle, Arthur’s youngest son, tried his hand at writing Sherlock Holmes stories after his dad died. I’ve read 'em - they’re not half bad. His best is a little better than his dad’s worst, I’d say.
In a lot of comic strips, someone in the family continues the writing and a hired gun does the art. Blondie and Bill Hoest’s Laugh Parade come to mind. Dik Browne’s children apparently do Round Robin on his old strips.
Jack Kirby’s kids have taken stabs at continuing the family business, but are not (creatively) involved with the current Kirby revival from Dynamite Comics. Alan Moore and R. Crumb both have daughters whose early comics efforts have been warmly received.
The Flying Wallendas of tightrope-walking fame went on from where Karl, the patriarch, left off.
THE Puccini represented the seventh generation of musicians in his family.
I guess those aren’t the same type of situation as just beginning in middle age when the parent dies, but I’m throwing them out there anyway.
But didn’t he have the help of John Dickson Carr?
True. Dunno how much, exactly.
I’m not sure it’s an art, but Racing seems to have a lot of this. Lots of jr’s in Nascar.
Open-wheel racing, too…in the starting grid for this year’s Indy 500, you have Marco Andretti (son of Michael, grandson of Mario), John Andretti (nephew of Mario, son of Aldo, who was also a racer), Tomas Scheckter (son of Jody), and Graham Rahal (son of Bobby).
This is a good one. (As an aside, old Looney Tunes cartoons are one of the primary ways kids today learn about the pop culture of the 1940s, and likely have been since cartoons from that era were first shown on TV.)
OK, I’m not familiar with racing but that might be the answer to my main question. This certainly sounds like a substantial number of examples.
Carving statues out of mountains in the Black Hills?
Mt Rushmore, started by Gutzon Borglum and “finished” by his son, Lincoln.
Crazy Horse, started by Korczak Ziółkowski, and continued by his children today.
Another notable example is Dwayne Johnson…his mother was the daughter of Peter Maivia, and his father was Rocky Johnson, both wrestlers. The names of the two were combined into his original WWF ring name, Rocky Maivia. (Interestingly, while the Rock was billed as a third-generation wrestler when he started out, Peter Maivia and Rocky Johnson debuted at about the same time, and Peter’s daughter wasn’t a wrestler, so, really, he’s second generation.)
The Anoa’i family, to which the Maivias are connected (Peter was a blood brother to the head of the family, therefore those that came after him are considered cousins), also have multiple generations of wrestlers in various branches. (Plus several following in the footsteps of their uncles and cousins, if not their fathers.)