Re: Genre switching sequels?
Arguably, the "Alien" series.
Alien was a horror story with the trappings of science fiction, while Aliens was more a war movie in a science fiction setting.
Alien 3, of course, was just crap.
Alien was Sci-Fi horror, so was Aliens. Or you might call both Sci Fi monster movies.
Terminator. Of course, you have to ignore the last two films for this to work, but everyone already does that anyway.
Terminator 1 was a horror movie
Terminator 2 was an action flick
T1 was a Sci Fi monster movie. T2 was also a Sci Fi monster movie (without the killing).
Oh, how about the Lou Grant character being spun off from the half-hour sitcom The Mary Tyler Moore Show to the hourlong drama Lou Grant?
Perfect example of what the OP was talking about.
Duckman: I'll never forget the last thing my father said to me...
Cornfed: "Careful son, I don't think the safety's on"?
Duckman: BEFORE THAT!!!