No. You would have to file with FL unemployment which would then get the records of your work history, just as PA did when you applied. If you do not put in a change of address, you could still collect, but that would be illegal.
You may need to check the laws in both areas (where you are and where you're going). I moved from Nevada to Oregon and they were aware of my move. My unemployment was being paid through the state of Nevada even though I was now an Oregon resident.
I will be moving to Florida from Tennessee. I am currently drawing unemployment. Can I file for unemployment in Fl. Online before I actually move?