Moving with NFA items is not real difficult there are certain things you need to do before moving. You need to apply for, and receive, permission from the ATF to transport your NFA items across state lines prior to the move. You do this by submitting a Form 5320.20. Let me say that one more time … you must have an approved Form 20 before you transport any of your NFA items (excepting suppressors and AOWs) across state lines.
And while suppressors and AOWs technically do not require a Form 20 to cross state lines, It is strongly recommend that, in the case of a permanent move, you submit a Form 20 for them in any case so that the NFRTR can be updated to reflect your new state of residence.
Finally … once you have your approved Form 20(s) in hand then you may move with confidence.
The above is also valid if you move residences within your current state.