Excuse the title, but I was trying to make a list of sequels that either follow the story but change genres? Or the reverse, I guess, would be a spiritual sequel that has no literal connection to the previous one, but carries on the themes and ideas.

The best example I can think of for the first type is Alien -> Aliens. Alien is your typical horror film with a medium sized cast being picked off one-by-one by the slasher. Aliens is an action flick with horror elements. Also of use here would be the Evil Dead --> Army of Darkness movies. Evil Dead 1 was a fairly straight, if a little quirky, horror flick. Evil Dead 2 was a reboot with a lot more silly thrown in. And Army of Darkness was pretty much a comedy first and a horror (? Mideavil fantasy?) film last.

Can you help me come up with others?

Irregular
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Alien -> Aliens
Evil Dead -> Army of Darkness
Flash Back -> Fade to Black (video game)

Spiritual Sequels
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Final Fantasy Series
System Shock -> BioShock
Blade Runner -> Soldier (screenwriters called it a "Sidequel")