Welcome to OCDO. Your questions, while not about OC, are valid for the NY sub-forum. I will try to cover the more important base. Others with more direct local knowledge will chime in later to add to what I have said and clarifying as necessary.
If you get your permit before you move, I don't see why you would have to get a new permit after relocating. Unless I am mistaken, once you have acquired the permit, it is good everywhere it is recognized - note that large cities may have more restrictive gun laws, particularly NYC. Sadly there is no state statute (reciprocity) preventing home rule by municipalities, but (except as noted above) the permit is otherwise good generally throughout the state of NY. Renewing the permit at its expiration might be a different matter if your new location has more limitations.
Here is the wording of the applicable law:
"A license to carry or possess a pistol or revolver, not otherwise limited as to place or time of possession, shall be effective throughout the state, except that the license shall not be valid within the city of New York unless a special permit granting validity is issued by the police commissioner of that city."
Also it is my understanding that you may apply for a permit not only where you live, but where you are employed or have a business. These terms may help you if any are applicable to your situation.