Assuming New Zealand follows a similar procedure to that of many/most other countries then , although you and the children would have to apply for your own visas, it is usually the case that dependants (married spouse and children) find it a matter of formality to obtain permission to accompany the residence/work visa holder. You would, however, likely have to obtain any work permit for yourself entirely in your own right (i.e. in effect separately). I know someone who accompanied her husband to New Zealand some years ago and found the system there obstructive with regard to her work permit, but that may have been a particular case and rules/procedures may have changed since.