Rebecca Minnock, 35, and her son Ethan have been missing from Somerset for more than two weeks.
Andrew Butt, the partner of Rebecca's mother Louise, was arrested on Wednesday and charged with contempt.
During a hearing at Bristol Crown Court Judge Stephen Wildblood said: "Mr Butt has told a string of lies."
The case centred around Butt's car - a black Ford Focus - which Judge Wildblood said the defendant had driven Miss Minnock and Ethan to Cheltenham in.
But the court heard the defendant had "continued to tell lies" after his arrest.
"The only reason is that he knows precisely where they went and that he took them to Cheltenham in his motorcar that morning [27 May]," Judge Wildblood added.
Butt is due back in court on Friday for sentencing.
The court also heard a black Ford Focus was seen at Asda, where Rebecca took money out of a cash machine on the same day.
At a previous hearing, Louise Minnock confirmed Butt owned a black Focus and stated that neither she nor her daughter were drivers.
Since their disappearance, Miss Minnock and Ethan have been seen in Cheltenham and Birmingham where the police effort is now believed to be focused.
Ethan has lived with his mother since his parents separated in February 2013.
On 27 May, a district judge ruled Ethan should live with father Roger Williams after finding Miss Minnock had made false allegations and obstructed contact between the father and son.
A child psychiatrist recommended Ethan should live with his father and have supervised contact with his mother - a view shared by a social worker and a guardian.
At the hearing earlier, Judge Wildblood stressed:  "I think it is important to put right, so the public can understand, the seriousness with which the court takes in approaching the task of ensuring that both parents, if at all possible, maintain an effective relationship with the child."
A man is facing jail after being convicted of contempt of court over the case of a woman missing with her son after a custody ruling against her.