If the city/county to which you're moving doesn't allow those items as a matter of law, then your 5320.20 will be refused.
If the sheriff there just isn't friendly towards NFA stuff, but there's no actual law against it, then it will probably get approved.
Some states which allow NFA items also require licenses or special taxes, so you'd have to fill out whatever forms and get approval from the state BEFORE ATF will approve your 5320.20. This will usually involve leaving the firearms in locked storage in your former state of residence while cutting through the legal red tape, until you can legally bring the NFA stuff to your residence in the new state.