It depends.
How big are the boxes?
beer crate or in the cubic meter size.
What's in the boxes? A
customs declaration will have to be made and if they take an interest clearance in NZ will have to happen.
Regarding the clearance. Goods will be held in bond until the clearance
documentation is completed and any
fees paid. You may or may not get a grace period to do this sometimes 7 days and then will pay storage/demurrage. Small items it's possible (or used to be) to do yourself but you will need to get a customer number from
customs and complete paperwork. A lot of this depends on if it's a small package via courier or post (who usually do this as part of the fee) or if it's a bigger consignment.
So smaller items courier or post. I would expect similar post/courier cost.
Larger items up to car size talk to GT Logistics in
Auckland or a similar company. They specialise in consolidating freight into containers and dealing with all the paperwork at NZ end (your goods will be shipped together with perhaps a car or two along with other packages. If they don't do directly they will
work with somebody they know/trust at the UK end. The
shipping cost is per cubic meter but there are lots of fixed costs, such as customs,
import fees, MAF inspections etc so you have to account for these and compare with courier or post.