A request by police to put a border alert on the pair was issued on the day they disappeared, a police spokesman said.
"If those passports, or people try to use passports associated with those names to leave the country, an alert is brought up on the screen."
No alert had come through, which meant it was unlikely they had left the country, he said.
"[But] until we can confirm what may or may not be in the car, we are still looking at all potential possibilities that may have happened."
Police had so far been unable to recover Mr Beckenridge's vehicle from Curio Bay in Southland because of rough weather and tidal conditions.
As soon as conditions allowed, police would attempt to retrieve the vehicle, the spokesman said.
Police divers had been able to perform a "partial search" of the station wagon, which showed there was nothing inside.
"But ... there's only so much you can see with a partial search."
A number of doors were missing from the vehicle, he said.
Police had received "a number" of calls from the public, but they had not resulted in strong leads, he said.
Mr Beckenridge is also known as John Robert Lundh, Knut Goran Roland Lundh or John Bradford.
Michael is of solid build and of Asian descent. Mr Beckenridge is 1.75m in height, of medium build and of Swedish descent.
Anyone with information that might assist in locating the pair should contact the Southern District Command Centre on 03 471 5002.
Information can also be provided anonymously to the Crimestoppers organisation on 0800 555 111.
Meanwhile, a Family Court judge has said the reason Michael was taken away from his stepfather was because Mr Beckenridge was alienating the youngster.
Judge Christina Cook would not release the full Beckenridge judgment, but said her reasons for not releasing it could be published.
The reasons clarified the proceedings were not related to domestic violence, but to the Care of Children Act.
She said a decision was made, following specialist advise, to change the care arrangements from Mr Beckenridge to Michael's mother "due to findings that Mike had been alienated in the care of Mr Beckenridge".
Mr Beckenridge had not participated fully in the court proceedings, Judge Cook said.
"My principal concern is that when Michael is located, the publication of the decision may create further stress and trauma for him."
Prime Minister John Key told reporters today he did not have specific advice on the case, but believed the alert system for the border was working well "for the most part".
"There are always exceptions and if there are loopholes that need to be tightened up then we'll look at that.
"It's never quite as straight forward because you're writing the law that has to apply to a specific circumstance and you're writing it on a general basis, but for the most part I'm happy that the law is working okay."
