The rise relates to Kiwi kids wrongfully taken overseas, rather than children brought here.
Otago University family law expert Professor Mark Henaghan said that could reflect the high cost of family law cases, and tight restrictions on legal aid.
"Many lawyers won't do legal aid work now because you get paid so little for it, and some people just can't afford to do it without legal aid."
He knew of one woman who approached 10 law firms, all of which refused to take her legal aid case.
For many, seeking a court order to settle custody or allow the children to go overseas would be too costly.
"You're not going to go to court and spend $15,000 to $20,000 on a lawyer to try to get a court order. You're just going to go."
Professor Henaghan said the convention aimed to deter parents from abducting children overseas if they lost custody. But today, fleeing parents were more likely to be mothers escaping family violence. "There's a growing literature now saying that a lot are leaving for good reasons and they've got no option but to leave."
While the convention includes exceptions where a parent or child faces a grave risk, in practice courts generally presume child protection and domestic violence laws in the home country will protect them.
District Courts general manager Tony Fisher said there wasn't enough information to know why the number of cases was rising.
Britain has also reported a stark rise in cross-border disputes, from 27 cases in 2007 to 253 in 2012.
The Office of International Family Justice said last year it dealt with four cases involving children from New Zealand, and one case related to a British child brought here.
Dad 'powerless' to get girl back
An Auckland man can't understand how his 14-year-old daughter was able to leave the country without parental consent and says he feels powerless to get her back.
Peter Sanders, 38, said his daughter Melissa flew to Brisbane last month by herself and was met at the airport by a person who is not, and has never been, a legal guardian of Melissa.
The young teenager left without either of her parents' permission.
Mr Sanders called police when he discovered she was gone and they alerted Australian authorities.
He said police were looking into the case, but had been told it was a "civil matter".
He is outraged Melissa was able to leave in the first place and wasn't stopped in Australia.
"They let her through, which I think is absolutely appalling.
"They're basically saying any child can just go overseas; that's what they're saying ... When I heard she was on a plane, I was pretty shocked."
Since Melissa left, he had spoken to her only once, through Facebook. He believed she was happy and had been enrolled in a school in Queensland.
Mr Sanders' father contacted MP Phil Twyford, who wrote to the Police Minister Anne Tolley asking for assistance. A spokesman for Ms Tolley said her office was waiting for advice from the police.
