FWIW, when I moved they did not issue a new permit to me with the new address. I just put my own sticker on the back with the new address similar to what the DMV does for CT driver's license. When I called DPS to report the new address, they were just interested in updating their database more than anything. When it came time for renewal, the new permit had the new address on it.
This was about 10 years ago, so I'm not sure if anything has changed.
If you have not done so already, make sure the notification is done within 48 hours after moving.
How do I change my address on my State Pistol Permit?
You can either call (860) 685-8290, or mail a letter to the Emergency Services and Public Protection, Division of State Police, Special Licensing and Firearms Unit, 1111 Country Club Road, Middletown, CT, 06457. Change of address is required within 48 hours, and the letter should include pistol permit number, name, and date of birth, old address, and new address.