Hi Imsee. Welcome to the forum.
There is a state pension in NZ that they refer to as 'superannuation'. You have to be a tax resident for 10 years to be able to claim it though.
Here's a bit about
private pensions that are outside NZ. 2 years ago new rules regarding the transfer of UK pension funds to overseas schemes came into force, resulting in NZ Superannuation Scheme providers having to get specific approval from the UK HMRC to be able to accept transfers, so you need to be very careful how you go about it if you want to transfer to NZ.
I don't know if you'll find something of interest in
here and
this is the Social Security Agreement between NZ and the UK if it's of any use. A bit about
superannuation .
Your husband is doing the right thing gathering all the information together. The better prepared you are, the less likely it will be that there will be any nasty shocks along the way.