Countries will extradite their citizens if they've been charged with a crime, but it has to be a crime in both countries, and it needs to be of some severity. Parking tickets aren't enough to get me extradited from Canada, even though not paying them is a misdemanor. Similarly, charging me with blasphemy in Iran and asking for me to extradited won't work either.
NZ needs to have made copyright infringement an indictable offence, and they need to have done so before Mr Dotcom was charged.
If not, and if they wish to get rid of him, they need to ensure somehow that he doesn't have a lawyer, and then hope he can't defend himself adequately.