What is this New Zealand insect and what is it in?
The quality of the pictures is not the best one, but due to the fact that this is NZ, it's probably a moth from the species Liothula omnivora.

It's a kiwi insect (that is, endemic to New Zealand). Its English common name is bagworm, and its Maori common name is pū a Raukatauri. The English name explains your question ("what is it in?"). According to Farm Forestry New Zealand:


  The common name of this insect refers to the strong silk bag made as a refuge by the caterpillar. The bag is brown to greyish, often covered with fragments of vegetation, and so tough it cannot be torn open. 


Here is a picture of it:



And here is the adult, for comparison: