Inspired by this thread. Examples of this may be pretty thin on the ground because crossovers themselves are fairly rare, but I’m thinking of something similar to the Bones/Sleepy Hollow crossover–Bones being a somewhat serious show taking place in the real world with no supernatural elements and Speepy Hollow being about magic. So shows of different genres, different bases of reality, or just very different “feels” to them.
“I Love Lucy” crossed over with “Superman” in such as way that children watching Lucy would see no reference to “George Reeves” (or any other indication that the Superman who showed up in the Ricardo’s apartment was not a Kryptonian).
A character from Warehouse 13 (a wacky supernatural comedy-adventure) turned up on Alphas (a much more serious superhero-drama).
Last season’s ScoobyNatural episode comes to mind…
I think my favorite is the crossover between Dirty Jobs and Sesame Street. Mike Rowe did Oscar’s job.
John Munch crossed over not only to other detective shows, but also to Arrested Development
Also, characters crossed over in both directions between Warehouse 13 and Eureka (a wacky pseudo-scientific comedy-adventure which was very explicit in every episode except the Warehouse 13 cross-over that the supernatural did not exist and all the weird occurrences on the show had rational, (pseudo) scientific explanations).
And The X-Files(!)
And the X-Files, in turn, crossed over to the Simpsons.
(I see that one as a crossover, not a parody, because the X-Files characters acted as themselves.)
I loved that episode, but…I don’t think it really counts for the purposes of this thread. Sam and Dean are transported to a magical alternate reality/pocket universe which is based on an episode of Scooby-Doo. In the type of cross-over (I think) the OP is referring to, a character from one show just shows up in a very different sort of show with no in-show explanation for the genre/mood jump - apparently two very different shows, such as Bones and Sleepy Hollow, are somehow set in the same world.
The campy, 1960s Adam West Batman crossed over with the much more serious The Green Hornet.
While not really cross-overs, Batman also featured a lot of cameos by characters from other ABC shows, regardless of genre or even time period (Col. Klink from Hogan’s Heroes, for example).
The drama Lou Grant was a spin-off of the comedy The Mary Tyler Moore Show.
The superhero genre as it currently exists is basically an example of this that has become a genre in itself.
Arguably, it’s always been, to an extent, as were its pulp predecessors, but when supers started gaining prominence again, the companies started drawing in the war, western, space adventure, sword and sorcery, horror, etc, characters into all one universe with their superheroes.
The Practice was a dark serious drama about defense lawyers. Ally McBeal was a light musical comedy drama about a law firm. Both were David E. Kelly series on ABC and FOX respectively that crossed over. It was a big deal because cross network cross overs were and are rare.
The Apprentice crossed over with House of Cards.
Some of the doctors from St. Eligius visited Sam Malone’s bar.