I have had many occasions where people have taken a UK registered aircraft to the USA and asked for an export C.of A. and I ask them why ?
There is no legal rerason for a country to issue an export C of A just because the aircraft is being exported. There is often confusion between commercial exportation ( i.e. Export licence etc) and Airworthiness Exports
A US manufactured aircraft ( Boeing Douglas Piper etc) does not need an Export C.of A from any country to go on FAA register.
In fact to go on FAA register a foreign built aircraft needs an Export C.of A. from the country of build - not register.
Some countries ( LIke Norway) do not have any such document as an Export C.of A.
CAA charge a full Cof A fee to issue an Export C.of A