New Zealand Law
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Under the Care of Children Act you must have consent of both parents to take a child from New Zealand, as I expect you know. If the father will not consent you may need to make an urgent application to the Family Court for permission. In the circumstances you describe, with a return air ticket, I expect you would get this, and quite possibly get costs if the father is shown to be unreasonable. Of course this law applies equally to him, and he cannot remove her from New Zealand either.
How old is your daughter? Have you been through FDR mediation as yet?
If you apply to court with that further information, he will be made to look unreasonable, and pay costs. So perhaps the next step is to tell him, you have complied with the conditions and that if he doesn't immediately consent, then you will apply for permission and seek full legal costs from him.
He cannot use the citizenship issue to stop the travel as it is irrelevant to the issue of whether you will return with the child.
have you instructed a family lawyer?
I mean if he won't consent then you can apply to the court for the courts permission. I suggested getting a lawyer because you would have good grounds for claiming the costs of the lawyer from him, as he is unreasonable. This sort of application is not simple, and I wouldn't recommend you do this yourself.
I am online. If you have a negotiated parenting agreement, he is entitled to reopen this. The first step would be another round of FDR mediation. If it was inappropriate to have further mediation, and the mediator agreed that mediation should not take place, then he could apply to court.
However a threat to your daughter that you would go to prison is completely wrong. He has no power to go to court to make you go to jail. The very threat is itself a good example of why he should not be permitted any further contact other than what already exists. It may even indicate that his existing contact should be supervised. Using the child to make threats against a parent is wrong.
He could apply for equal parenting. I would suspect that if he is not paying child support, then he is not really committed to the idea but is saying he wants equal parenting because he wants to avoid paying his share of child support. But his behaviour and the threats indicate that he is not suitable.