I did my conversion the other way, ie NZ to Aus, it should be the same process either way. It was very simple, I went in to the CASA office in Darwin and they did it on the spot (though I had to make an appointment so she could set asside enough time). The actual licence took a couple of weeks to arrive but she said I could have a temporary licence immediately if I needed to fly before that, I didn't so I didn't worry about it.
I used my NZ Class 1 until it expired. I did have trouble when I did my MECIR in Aus as CASA wouldn't accept my NZ Class 1 despite having issued me the CPL in the first place. Lucky I had a piece of paper from the CASA medical bloke that clearly stated that my NZ Class 1 was valid till expirey in Aus.
The only trouble I had was that my aerobatic rating didin't carry over (as it's a logbook endorsement rather than a rating). Also the only retractable I'd flown (T6 Harvard) didn't appear in the CASA list of retractable undercarriage aircraft and the woman wasn't going to give me the endorsement until some other guy in the office said he knew of the type and it was ok.
Beware that there is no such thing as a co-pilot instrument rating in NZ so if that's all you have then you won't be able to convert it to a NZ instrument rating.