Has a series ever actually successfully passed the torch?

Yes, she did

How about Rudy Huxtable passing the torch to Olivia on the Cosby Show to fulfill the role of a younger child in the Huxtable house?
I was going to say Cheers did it similarly with Coach/Woody and Diane/Rebecca but they never appeared on the show together.

I could have sworn there was an episode in which they both appeared and interacted, but it has been a long time since I watched it.

I’m not really sure if this qualifies, since it wasn’t a clear, singular handover, but some of episodes in the later seasons of Murder, She Wrote have Jessica Fletcher absent altogether (Murder, She Missed, if you will), with other characters taking the investigative lead.

The Thick of It didn’t suffer terribly when it replaced Chris Langham with Rebecca Front in the “bumbling but well-meaning government official” role halfway through its run.

In Unforgotten, another UK mystery series, Sanjeev Bhaskar as “Sunny” Kahn was the assistant to Cassie Stuart, played by Nicola Walker. She left the series and Sinead Keenan as Jessica James replaced her, but Sunny was given a more prominent role.

Sandi Toksvig made 16 guest appearances on QI before taking over from Stephen Fry as host.

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Why the Bourne series?

I mean, he made one Jason Bourne movie. Not successfully and not with Matt Damon passing the torch.

Someone had to run the filling station.

How long ago did they call gas stations, a filling station?

I’m confused because I haven’t seen or read any of the works mentioned in the OP, except for the Bourne Series. Aren’t the new characters supposed to have shared screen time with the one they’'re replacing?

Yes, that’s why this one doesn’t work.

Yeah. Shared some scenes, then eventually took over for that person, and the original person leaves…all on the same show.

I was surprised that, when Kyra Sedgewick left The Closer, a lesser-known character would just be promoted into her role and the series would continue as Major Crimes.

In The World Is Not Enough, John Cleese’s character is introduced as Q’s assistant. (Bond sarcastically refers to him as “R”.)

In Die Another Day, the old Q was gone (Desmond Llewelyn had died), and John Cleese’s character was the new Quartermaster.

Might not count, but how about Maverick? Jack Kelly basically took over the title role when James Garner left after season 3 (although he did reappear in several episodes later on)

How about Happy Days - that show about Richie Cunningham? He and Fonzie were on the screen together all the time - and the Fonz, of course, led the show after Richie left

Would Radar leaving MASH and Klinger taking on the role of company clerk qualify?

And Sinclair reappeared twice after leaving the show. OK, once was just on just on video, but the other was a two-part episode.

Or Jack McCoy taking over the role of DA?

Oh, and Mel Gibson pretty much had the torch passed to him by James Garner in the movie version. Now, it ultimately does turn out that Jack Garner is playing the OG Maverick and Mel is his son, but it’s the same idea.

They didn’t make sequels, but it is a great movie.