Ten have been handed over and the MoD said it planned to transfer the remaining 82 from Camp Bastion “as soon as possible”.
However, legal challenges prevent the transfer of seven detainees.
On Thursday, two men dropped challenges to their detention and can now be transferred, the High Court heard.
The court confirmed that, subject to detainees confirming that they did not want legal representation, they could be transferred to the Afghan National Detention Facility, within the US Bagram airbase in Parwan province.
Defence Secretary Philip Hammond said he was “very pleased” that the High Court had agreed to vary the ruling that blocked the transfer of the suspected insurgents, who are being held at Camp Bastion without charge.
He said: “Terms have been agreed that will allow those detainees who have requested transfer to Afghan custody to be handed over.
“Unfortunately, the injunction remains in place for some detainees who are still represented by human rights lawyers.”
Mr Hammond said UK forces had given the detainees telephone access to UK lawyers.
‘Minded to withdraw’
On Thursday, the two main plaintiffs cases in a High Court hearing in London due to start on 19 July dropped their challenge.
The High Court was told that Niahmutullah Haqim and Mohammed Ismail were “happy to go”.
Seven other detainees involved in the hearing are to be contacted by their UK lawyers shortly to see whether they want to continue their legal challenge.
The judge, Sir John Thomas, was told that another applicant – a Mr Yahyah – was also “minded to withdraw” from next month’s High Court hearing.
James Eadie QC, appearing for the Ministry of Defence, told the judge the case seemed “on the verge of disappearing”.
On Thursday night, the first 10 detainees were put on a plane to Parwan within an hour of confirming they wished to be transferred, the MoD said.
The group is the first to be moved to an Afghan facility at Bagram airbase, about 700km (450 miles) from Camp Bastion. It is monitored by US forces, and the Ministry of Defence considers there is no risk of mistreatment.
The transfer of detainees from the UK’s main base in Afghanistan to Afghan authorities was banned in November by Mr Hammond because of fears that they would be abused.
However, earlier this month he said that the facility at Parwan had received “positive reports” from humanitarian organisations.
The MoD’s plan to begin the transfer this week prompted lawyers acting for some of the detainees to apply successfully for an injunction from the High Court.
Many of those being held are suspected of involvement in preparing, laying, or facilitating the use of roadside bombs against British forces, or have been picked up at the scenes of shootings of British soldiers.