For down to earth series, there’s no need to wait for the Next Generation - just pick up the thread.
Movies, of course often reprise roles, sometimes with much of the same dialog as the original.
They do this switching a British series like The Office to a new cast. I think they even kept much of the scripts.
So why not combine those two tricks and pick up an American TV winner and give it a new cast.
Cheers, with simply new faces. Bartender still named Sam, postman named Cliff, bar regular named Norm. Maybe new scripts, but not necessarily.
Friends with new twenty somethings in their first jobs, ignoring the baggage of the later years’ story lines.
Or, à la CSI, Friends-Miami and Friends-Las Vegas
Of course, I think Lucile Ball did this a couple of times, where she sort of picked up her old role, with just enough changes to cut out any former writers/partners.
“I Love Lucy” …as Lucy Ricardo" “The Lucy Show” …as Lucy Carmichael" “Here’s Lucy” …as Lucille Carter" With Vivian Vance playing differently named sidekick characters.
Any other scenarios you could suggest for redoing a past TV series?
They’ve tried this sort of thing – they had a “Bonanza – the Next Generation” and Mrs. Columbo, neither of which was a big success. They’ve had more luck with sitcom spinoffs, often in very long threads: Danny Thomas > The Andy Griffith Show > Gomer Pyle USMC and Mayberry RFD
All in the Family > Maude and The Jeffersons and a bunch others.
And the whole “Happy Days” chain, and the "Mary Tyler Moore chain.
Then there was the Prime Time Soaps spinoff series – Dallas and Falconcrest and that stuff.
The only genre-crossing examples I can think of are Mary Tyler Moore > Lou Grant
No one has mentioned Man from U.N.C.L.E. which begat Girl from U.N.C.L.E and NCIS (well when someone was asked to discribe Ducky - They said he looked like an older Illya Kuryakin - and you know, he does)
It also spun off a movie and God only knows how many toys and even a cologne if I remember correctly.
In a weird sort of way, I suppose you could argue that the Warner Westerns of the late '50s and early '60s would qualify.
Maverick, Sugerfoot, Chyenne, and two or three others had an Old West that basically functioned as their own universe with characters that traveled within the family of shows. My favorite was Adam West who would be Doc Holiday on each of the different shows in the span of a couple of months or so.
I agree with **Push You Down ** that the thread title and OP contents don’t mesh.
That said, in my view Cheers and Frasier qualify to have been a franchise, albeit limited to two, but pretty much exactly like the original Star Trek, Next Generation, et al franchise. The characters inhabited the same universe, and many of the characters from Cheers appeared on Frasier at least once. This included: Woody Harrelson, Ted Danson, Shelley Long, George Wendt, John Ratzenberger, Rhea Perlman, and of course Bebe Neuwirth several times. Explanations were given about some of the Cheers “barflies” when other characters appeared in Seattle.
And if you want to go with “interlocking” series where characters are appearing on other parallel shows, the Law and Order franchise is most definitely a franchise.
Petticoat Junction and Green Acres were a franchise, although GA wound up being more of a Bizzaro World viewpoint.
By the end of its run My Three Sons was something of a franchise with Steve, Robbie, Chip and Ernie involved in their own stories, and relatively little whole-family hijinks.
And there have been a plethora of series with “The New” in their titles. Sadly, as fans from shows as varied as *The Monkees, WKRP * and The Odd Couple discovered, lightning seldom strikes twice in the same setting.