^Well, if you want to go Shakespearean, the Falstaff character from the Henry IV history plays got his own comedy spinoff, The Merry Wives of Windsor.
^Well, I don't know if a tonal shift is really a genre shift. But yeah, the original trilogy was just a fun homage to Flash Gordon and Saturday matinee adventure serials, while the prequel trilogy was an attempt at a serious historical-political epic with a message about the abuse of power.
From Dusk Till Dawn managed it in a single movie, switching from a criminal chic flick to a horror film half way through.
How about the Star Wars Holiday Special? That's a genre shift, isn't it? ETA: And in the related topic of genre-switching TV spin-offs: M*A*S*H and Trapper John, M.D.
Technically, Trapper John, MD is considered a spinoff of the movie M*A*S*H rather than the TV sitcom -- although that was just done to avoid paying royalties to the sitcom's producers. But yeah, I guess both shows derived from the movie ended up going in a more dramatic direction, though in the case of M*A*S*H the series, it was a more gradual transition.
To be fair, it was never meant to be Cruel Intentions 2, but a TV pilot that was rebranded Cruel Intentions with the hope of getting some more money.
Not a sequel, but the Dukes of Hazzard prequel to the theatrical movie became a college style romp, in effect totally different in style and tone to the rest of the series and TV show.