What happens in Florida when the power of attorney for someone who has dementia dies?

I can't speak for Florida specifically, but generally if the wording in the POA identifies a successor, or someone to act in an "either/or capacity" with the person who's just died, that alternate person would then become the proxy. Or were you referring to the person who created the POA as the person who died?

I'm trying to find out what happens when the power of attorney for the person does before them

If you are the named standby guardian, you present the original POA document and the death certificate of the first person to prove you are taking over.