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There is no legal requirement to notify for interstate transport of suppressors.
Most people use a Form 5 with the box ticked to indicate a permanent move, though - the address is right there on the F5.
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The form 5 is used for tax exempt transfers of NFA items, not for moves.
In the past, you used to be able to send a letter to the ATF for a move. In recent years though, the ATF has stated that they want the 5320.20 instead. While you are not required to get approval to move the silencers out of state, it is recommended to report the move and keep the NFRTR up to date. If not, if may cause problems later on should the silencer need to be transferred or serviced as the address listed will not match.
I sent in a couple 5320.20 forms for my old cans back in February, took about 3 weeks in total from the time i emailed them to the time they arrived in my mailbox.