Based on the two sets of 5320 forms I sent in, ATF seems to want to send them to the address you're going to, not the address you're leaving from, which makes zero sense.
The second time, in the box where the form asks you where you want them sent to if different than your current address, I put in my current address. And they still sent them to the new address. I only ever got them because the people I was buying the house from got the forms in the mail and saved them for me.
So if you fill out a 5320 and never get it back, it may have gotten sent to the address you list as moving or traveling to, which may not be one where you'll ever get it, if it's a training center or some such.
For the 5320 and move aspect, nothing seems to rile people up as much as this aspect, with insults and name calling aplenty.
Factual data:
Your are not legally required to obtain permission via ATF to move a suppressor across state lines.
ATF states on the back of the Form 4 for you to let them know of a permanent change of address.
Now people run the gamut from filling out a 5320 for suppressors, to doing nothing, and seem to take umbrage if someone else does things different.
My thoughts, and what I did:
I'm not filling out a form for permission from ATF to move a suppressor if it's not legally required. 1. No reason to ask for permission if not required 2. With only a silly 3 lines per form, it' would take a bunch more forms to fill out and mail in /get back.
After the move, I did write a business letter to ATF listing the serial numbers of the suppressor I moved, and what their new permanent address was. 1. It says to do that on the form, and it would be less than ideal to have to prove to a jury why I wasn't required to do that, in the rare event something untold happened. 2. As stated above, if I ever transfer the suppressors, ATF could if they wished give some pushback as to the fact the address on the Form 4 wasn't the same as what they had in their register from the Form 4 from when I got it. You could probably say you forgot / lost in the mail / etc., but why bother with all that when you can mail in a letter for 46 cents?
After month or so I got a letter back from ATF with my letter and a Received or Filed or some such stamp on it, so I'm set.