Is there a word for when key signatures change every few bars, similar to how time signatures can change every few bars -> "multi-metric"?
Changing key signatures/keys but NOT SIMULTANEOUSLY
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There's no specific term for the fact of changing key signatures frequently (but see ViviRukisha's answer), but in light of the term multi-metric I see no reason not to coin the term multi-tonal or multi-modulatory. The latter might be better to avoid confusion with polytonal.
Yes. This is often referred to as Modulation. However, it is not mandatory for a change in key signature when it comes to modulation. This can be achieved by the use of accidentals as well.
No there isn't, and I counsel against inventing one! The distinction between 'multi', 'poly' or any other prefix you could come up with won't be immediately clear, and there will be confusion between your intended meaning and SIMULTANEOUSLY multiple keys.
If you want to describe this, do it in plain English. Something like 'This passage has frequent changes of key signature'.
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