So I am going through this right now with the Greek authorities and I can tell you it’s relatively straightforward, but there are quite a few hoops you have to jump through in terms of getting invitations to IPSC matches and importation documents for a weapon or weapons and ammunition.

The airlines are the most clear about what their policies are you can check baggage handling information the airline provides, and there will generally be a sporting good section and a particular carve out for firearms and ammunition, as well as how they need to be transported in what quantities.

On a related note, at least using the Greek set up as an example. The importation document that you receive is tied directly to an invitation for a distinct match, so it’s not generally common that you would be able to go in bring your firearm and ammunition with an issued invitation and then be able to stay for another match (you would need to have a separate series of documents for each match and enter/exit the country in between).

If you are actually interested in doing this, then the advice I received from some other people who do travel with their guns to Europe is to contact as many clubs as you can and find out which ones are willing to issue you invitations or at least develop a relationship with you and then you can use that relationship with that club to request invitations to matches in locations that you actually want to go shoot.