I'm going through the same thing right now. Have you ever traveled with your cats before? I highly recommend a mild sedative, but not one that'll knock them out. You don't want to have to worry about whether or not they're still breathing. But a light sedative will keep them calm and less anxious.

Littlewing where are you living in the states right now....I hope you realise that the process is about 6 months long getting your pet up to the standards. If you are in Cali, I could help you with a pet relocator. I am not sure if you can put all the cats in one big box or if you have to do it all separately, but can purchace the boxes vertually anywhere like Petco etc. But that is the least of your worries......www.biosecurity.govt.nz/import...spe.htm
There is the list of what you have to do, and what forms to have, and what forms to fill out. Make sure you start this EARLY, or else you will end up leaving your pets behind in your rush to get here. (I did that and regret it). It also cost A LOT of money. It was going to be $1,500 for my dog. So you with 3 cats could be a lot lot more. There are 2 ways of doing this.....by yourself reading the forms from the link I just sent you, or get a pet transporter company to help you. They make sure everything happens right, which is important. These pet relocators also have a list of vets that they work with, but if you do this on your own make sure that you ask the vet if they have done pet export to New Zealand. Very important because New Zealand is the most strict country in the world. One other side note, wait for your pet to leave and dont leave your pet with friends that will take care of the rest because you cant rely on other people (Again something learned from experience).Let me know if I can help you further.