You MUST Notify when you move, per MGL. Failure to do so allows revocation of your LTC.
Even though you move, ONLY the town that issued your permit can revoke/suspend. The new town can NOT modify the terms of your permit. Thus, unless you do something bad, your permit will be good until it expires.
Every renewal is treated like a "new applicant" and at that time, the chief can deny, issue, impose any restrictions he wants, downgrade you to anything that he wants. There is an appeals process thru the courts, but they almost always look at a "permit is a permit" and won't interfere with a chief's discretion (even if it is a downgrade from LTC-A to LTC-B).