Do buffs affect your party or only the character you're using?
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I just want to know: if a character who ate the buff dies, do I have to feed the next character in the party the same buffs?
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If you are talking about party wide buff you get from foods like reduced stamina consumption, increased element damage / resistance and so on those are considered party buff not specific to a character. You can easily test: use a potion and then jump off a cliff. The buff icon should still be there after you are switched to the next character.