It's all incorrect, actually. If you find a shipper, you are good to go. But it's a huge "if".
In practice, any firearms be they air or powder nowadays will almost certainly need to be freighted by a carrier that accepts them. This is fine if you have 30, 40 pieces, but ludicrously expensive to pointlessness if you don't.
Regular courier companies just don't want to know. However do take a look at Parcelforce's site as recently IIRC Garvin pointed out some interesting info in this regard that was published there, and good luck.
Customs are not interested if the airgun/s are sub-12 ft/lbs, otherwise an export licence is needed, again not a big deal, been there, done that.
Customs in the US, you will need to find out for yourself.
Personal export i.e. as hold luggage is easy, obviously if you conveniently happen to be going that way anyway.
**WANTED**: WEBLEY PATRIOT MUZZLE END; Any Diana/Original mod.50 parts, especially OPEN SIGHTS