The standard procedure is documented
here. In a nutshell: you complete the INZ 1023 form and send that with both your old and new passports to INZ. Certified copies will do if you want an eVisa linked to your new passport, but you'll have to provide the original if you want a physical label (for reasons that I hope are obvious).
I do not know how the lockdowns have impacted the processing of such applications; your best bet is to ask INZ directly. Before the recent lockdowns, this was a very quick process (usually within days, and no more than 2 weeks in 90% of the cases). What I
can tell you is that your application may well be redirected to another branch: I had a new visa label issued in June 2021 which was handled by the Palmerston North branch.