How can I legally send a gun to the USA?

Q: I have three shotguns that 
once belonged to my father. 
I hold the licence for them, but wonder if you could tell me how 
I would go about sending them 
to my brother in the US?

Send a gun to the USA

A: If you are a shotgun certificate holder, exporting a gun to the 
US is straightforward because certificate holders may export shotguns appearing on their certificates as their personal effects without an export licence. 
The Home Office Guide on Firearms Licensing Law (2016) deals with this 
in chapter 28. You can see the full guide here.

The guidance states:

28.31 Export licences are not usually required in the following circumstance. Holders of valid firearm or shotgun certificates or visitors’ firearm or shotgun permits may take abroad with them, or have sent, as part of their personal effects, without an export licence, any firearms, shotguns or related ammunition entered on the certificates if the certificates are presented by the holder, or their 
duly authorised agent, with the 
firearms and ammunition to the 
officer of the Border Force at the 
place of exportation. The exception also includes telescopic sights.

28.32 From time to time this exemption will not apply to firearms consigned to certain destinations. Further details on the UK export control licensing system, including guidance on the export of firearms, may be obtained from the Gov UK website, prior to export. See the specific BIS firearms guidance here.

If you need an export licence for the country of destination, you have to apply to BIS via the SPIRE export licensing system. It is worth asking one of the bigger auctioneers, which are likely to export guns from the UK, whether they would do this for you. They would charge a fee but this might be easier than doing it yourself.