According to Queensland news reports, Barnett initially fled to South Africa, where she married Juan Geldenhuys in 1995 and later gave birth to a son, now 17, before moving to New Zealand and becoming a citizen.
She and her children had been living on the Sunshine Coast for at least the past four years, the Daily Mail reported.
Australian federal police arrested the 53-year-old on November 4 and charged with Barnett was charged with international parental kidnapping and two counts of false statement in a passport application.
It was expected she would be extradited to the US to face the charges.
Her former husband and Savanna's father, Benjamin Harris Todd, told the Daily Mail he divorced Barnett citing "physical cruelty", which allegedly began shortly after they were married.
Mr Todd alleged Barnett would attack him while driving in their car.
He said she wouldn't sleep for "seven or eight days at a stretch" and would beat her head against a wall for what seemed like hours.
Barnett was diagnosed with manic depression but refused to get treatment.
Mr Todd was granted sole custody, with Barnett to have two supervised weekend visits a month.
It one of her visits, on April 22, 1994, Barnett and Savanna disappeared without a trace, the Daily Mail reported.
It was believed Barnett enlisted the help of Faye Yager, owner of an organisation called Children of the Underground, which helped women escape abusive relationships.
Savanna, now 20, who has been living under the name Samantha, supported her mother during a court appearance on Wednesday, where Barnett was denied bailed.
The Australian reported Savanna held up a sign during the hearing saying "We love you Mom".
Barnett's barrister, Jeff Hunter QC, said his client was not a flight risk and had a close and special relationship with the Sunshine Coast community.
In an affidavit tendered to the court, Savanna described Barnett as "an amazing woman, having raised both myself and my brother and always giving us the support we needed to become the people that we are".
