Done it several times - but as a transit passenger however the process is the same. You must have a customs clearance to enter Dutch airspace, the good news is that you can do it online but do not leave it to the last minute as they can take up to a month to issue although all of mine have been sorted within 10 days.
This is the link;
Application Consent for weapons and ammunition with an Annex to question 7 (Model WM 18 c)
You will need copies of the following attached to your application email
- Passport
- FAC (acts as authority to possess and UK hunting licence)
- Flight details
- Hunting invitation for country of destination
If you are with KLM you buy your ticket etc then a day or two later you have to phone and get authority to carry the rifle and ammo. Remember KLM/Schipol are notorious for missing connections, the official transit time is about 1.15 hours, if you rely on that the rifles will not follow you, ideally go for a transit of about 4-5 hours, its a pain but there is a better chance of both arriving at the destination. Last time thru on the way back with 6 hour transit time they only delivered my ammo , the rifle came two days later after me and Border Force raising a massive row with KLM.
KLM are now only flying firearms from a limited number of UK airports - check first, if its a KLM subsidiary then no chance. You may ask why I still use them - I can go from Newcastle or Teesside rather than travelling to Heathrow/ Gatwick.. simples.
Personally to Hungary I would go Easyjet from Gatwick - much simpler and they are the best organised of all the Airlines for firearms.
Good Luck !!