We had a nightmare recently, we called immigration to ask if we could have a replacement visa label inserted in a new passport and they said we didn't need to because it's all electronic now. Never doubted it as we are now in the 21st century and all but when we got to the airport they told us they needed to see the label and we should have brought our old passport, we had to wait whilst they called Canberra to verify our visa status, must say they were very snotty about it and had my young son in tears when they said he may not be able to travel.
So like you say, just make sure you have your old passport with you with the visa label in just in case.