animals do get exported from NZ from time to time (as with Australia, they seem to like to do random things to keep people on their toes....).
Five kiwi were sent over to the USA in 2010 (destined for both USA and Europe):
http://www.zoochat.com/17/kiwi-oe-147270/
Tuatara get sent out irregularly as well to select zoos (the San Diego Brothers Islands tuatara are a good example of something you wouldn't expect).
Morepork are unlikely though. I'd myself like more kaka sent overseas (they can be bred easily enough so surplus over-represented stock wouldn't be hard to breed on demand, as it were, for export). The NZ pigeon is another one that would be nice to see in overseas zoos that specialise in birds, but not many places breed them (the ones in zoos here are mostly rescue birds) so that's not going to happen. Weka would be a very good bird to establish overseas.
However I don't know if there would be enough interest in rather boring (to the public) NZ birds to keep overseas populations going even if there were some exports.