Fla. will certainly recognise a nonresident permit they issue. As will many other states.
I believe you will find that some states however, that would recognise say, the Fla. issued permit of a Fla. resident, (for example), may not recognise the Fla. issued "nonresident" permit.
Before I travel I always go online to the areas in which I intend to travel and check for their laws, and idosyncrasies concerning such. Laws that may have changed since my last check of them.
Ohio comes imeadiatly to mind as having some very strange rules. Such as no cary in any gov. owned building, such as a hiway rest area, (but we all know those are safe any way). When in a motor vehicle you wepon must be holstered, on your person, but visible from the outside of the vehicle. Each little town has it's own laws concerning carry. Some of these I think just changed in the last election cycle, but I'm not sure as I have not checked them yet.
Point being I guess, before you spend the time and money for finger prints, permit, etc. make an official check to see if this particular permit will fill your exact needs.