TV shows that switched networks during original run

Very rarely. Some variety acts switched from BBC to ITV in the 70s and 80s, Morecambe and Wise being the big example, MBB is the only fictional show I can think of going the other way.

The Goodies went BBC to ITV and failed badly as ITV hadnt realised how much their special effects budget cost.

The BBC had a great reputation under John Howard Davies then John Lloyd of developing great comedy for about 50 years. Panel shows seem to be the order of the day these days though.

 

The Danny Thomas Show
The Joey Bishop Show
The Real McCoys
Wagon Train
Bachelor Father aired on all 3 networks

 

Sorry, two different shows. Letterman actually couldn't take some material with him to CBS or they had to change the names. Larry "Bud" Melman couldn't be used, they had to use the actor's name Calvert DeForest. Viewer Mail had to become TV Mailbag and so on.

Besides, both shows have continued in the same timeslots with other hosts. Late Night with Conan O'Brien, Jimmy Fallon, and Seth Myers. Late Show with Stephen Colbert. It's the same way for the Tonight Show with Paar, Carson, Leno, Fallon. It's all the same show, even if they changed the host, style of comedy, supporting cast. (I don't know if Steve Allen counts to the Tonight Show, or Tonight! America After Dark, that's more debatable.)

 

The Orville (Fox to Hulu)
Daytime Wheel of Fortune (NBC to CBS, then back to NBC before being canceled for good)
Tiny Toon Adventures (first-run syndication to Fox)
Animaniacs (Fox to The WB)
JAG (NBC to CBS)
Matlock (NBC to ABC)
In the Heat of the Night (NBC to CBS)
The Jeff Foxworthy Show (ABC to NBC)

 

I'll Fly Away ran for 2 years on NBC before it was cancelled. PBS bought the rights to rebroadcast the series (plus added some material after each show) then produced the TV movie I'll Fly Away: Then And Now which served as a series finale. Although not quite the same as "switching" networks it was a unique situation. All the cast members returned for the PBS movie except for Jeremy London who had a scheduling conflict - his twin brother Jason filled in for him.

 

Brooklyn Nine-Nine (Andy Samberg). First 5 seasons on Fox then cancelled. Picked up by NBC.

 

When Fox cancelled American Dad! back in 2013, TBS stepped in and began airing new, first-run episodes of the series the following year in addition to the syndication reruns.