I have personally imported 4 shotguns over the years for personal use. On one occasion, I had Neil Anderson in Aberdeen, Scottland export a gun from his business to British Sporting Arms (Charles Schnabel). Upon arrival at BSA, the gun was shipped to my FFL holder.
The last time I imported from the UK, I carried back a 1886 H&H Rook rifle that was converted to .410. I put the gun on my British Shotgun Certificate and carried it to the airport. As this was an antique (Pre 1898), I asked H&H in London for an affidavit stating the year of manufacture with the serial number. On my arrival in the US, I showed customs the affidavit from H&H and they let me carry it in. No tax as it was an antique. Although I had the Bill of Sale, it was not required... easy peasey.
I have also used an importer in my state of residence to process the importation paperwork for two late model shotguns. I received the import permit in about 6 weeks. Again, both of these guns were on my British Shotgun Certificate. I took both the airport and checked them in. The airline looked at the Shotgun Certificate but were not interested in the US importation permit. Customs checked and logged the serial numbers, make and model. On arrival in the US, I picked up the baggage and went to customs to show the importation permit and pay duty. From there it was out the door...
I hope this helps.