You will find traveling with a firearm (hunting type) is very easy if you comply with the requirements of all the different agencies involved.
First contact your own RCMP firearm division. You may need a permit to export then import your firearm. Next go on line to New Zealand Police and click on firearms. It seems quite straight forward from there, you apply for a Visitors Firearms Licence (T licence) and apply to the Airport Police for a Permit to Import. Click on approved for import.
Traveling on an aircraft you must have the firearm (bolt removed) in one piece of checked (hold) luggage, and your bolt and ammo in a secound piece of luggage. I would recommend a hard case for your rifle.
I travel Air NZs direct 14hr flight Auckland- Vancouver and this does away with going through LA and US Customs.
Air NZ requested this year that the ammo be in a lockable container.
I bought a light steel cash box which took a 7mm mag bolt and 30 rounds in the plastic holders.
At Auckland your rifle will be delivered to the Airport Police next to luggage pickup and in Vancouver will come out with the large and fragiles.
Photo copy all relevant paper work and keep separate.
If you want to shoot a deer in NZ, I'll give you a spot.
Anywhere in the eye.
Good luck.