If you move within the same state there is generally not an issue other than being inconvenienced in returning to the gun store to pick up your item when the tax stamp is processed.
If you are moving to another state there are different situations. This discussion will talk about tax stamps involving a Gun Trust and not individual ownership.
1) if you move to a state where the item is not legal, a Gun Trust will not allow you to take that item across state lines.
2) If you move to a state where it is legal, you may not have the proper credentials to complete the purchase and pick up the item in the previous state. Many items require that you be a current resident of the state to make the purchase. If you are buying one of these items, and have a co-trustee who is not moving, they may be able to pickup the item for you.
3) If you do not have a co-trustee who will be able to pick up the item, you should contact the ATF at 304-616-4500, provide the serial number on your Form 4 and explain the situation to them. In some cases they have been known to permit a dealer to dealer transfer and a substitution of Form 4's but you should not count on this. Generally in such a case, you might have to cancel the transfer and start over again after it is properly transferred to a Class 3 dealer in your new state of residence.
4) Some items do not require you to be a resident so you may still be able to complete the transaction.
When you are ready to move the items to your new state, you will have to submit an ATF Form 5320.20 to update the registry with your new address. This process typically takes 2 weeks and it is the same form you would use to travel across state lines and return.