British nationals have left Gaza for the first time since war with Israel broke out last month.
The UK Foreign Office confirmed an unspecified number of UK passport holders had been able to leave via the Rafah crossing into Egypt on Wednesday.
It said the route was being opened for "controlled and time-limited periods" to allow some foreign nationals and injured Palestinians to leave.
About 200 British nationals are believed to be in Gaza.
It emerged on Wednesday that some 500 people a day would be allowed through the crossing, which is controlled by Egyptian authorities.
Thousands gathered at the border this morning hoping to leave, but it emerged that only those whose names appeared on a limited list agreed by the Egyptian and Israeli governments would be permitted to cross.
The UK Foreign Office said it had handed over the names of people who wished to leave Gaza.
On Wednesday evening, it confirmed some Britons were among a group of around 400 foreign nationals and injured Palestinians who had crossed, but did not say who or how many.
Earlier, the BBC spoke to British-Palestinian doctor Abdelkader Hammad, who was told he was in the first group allowed to leave.
But when he arrived at the crossing point, he found the route was closed and described frustration and confusion at the border.
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Speaking to BBC Radio 4's PM programme at around 16:00 GMT (18:00 in Gaza) from the crossing point, Dr Hammad said: "It's a little frustrating. We don't know what's going on...we don't know when the next group will go - if it will be tonight or tomorrow.
"It's dark - I'm not sure it will happen tonight, we'll see what happens tomorrow."
Speaking from the Rafah crossing at around 13:00 GMT, BBC News reporter Rushdi Abualouf said thousands of people were already at the border when it emerged only those on the list would be allowed through.
With no passport control or electronic ID system in place, the process is slowed by the need for an official to manually check the identities of every person leaving, he reported.
He also saw between 20 and 30 ambulances passing through the crossing carrying injured people into Egypt for medical treatment.
Routes in and out of Gaza have been closed since Hamas - which is proscribed as a terrorist group in the UK - attacked Israel on 7 October, killing more than 1,400 people and taking at least 239 hostage.
BBC News understands that 14 British nationals were among those killed. Three more are missing.
The Hamas-run health ministry in Gaza says more than 8,700 people have been killed since Israel launched air strikes as part of a military response to the attacks.
The partial opening of the Rafah crossing follows international diplomatic efforts to convince Egypt to allow people to leave and aid to be transported into the enclave.Rishi Sunak held a further call with Egyptian President Abdel Fattah El-Sisi on Wednesday evening.
Speaking to the BBC at the AI summit in London earlier on Wednesday, Mr Sunak said the government was committed to getting humanitarian supplies into Gaza, and helping UK passport holders leave.
He continued: "We're playing an active role in getting aid into Gaza to help those people who need it, but also diplomatically working with everyone in the region to find ways to move our British nationals out of Gaza and hopefully bring them home."
Foreign Secretary James Cleverley called the first departures from Gaza a "hugely important first step". He earlier said British officials are on the ground in Egypt "ready to assist British nationals as soon as they are able to leave".
Western officials told the BBC a team had been deployed to Arish, a city some 25 miles (41km) away from Rafah, to "ensure we can provide the necessary medical, consular and administrative support needed" for British nationals.
Among the British nationals in Gaza are Scottish First Minister Humza Yousaf's in-laws. He welcomed the opening of the border but said his wife's parents remained trapped without clean drinking water and rapidly diminishing supplies.
Both Mr Sunak and Labour leader Sir Keir Starmer have called for humanitarian "pauses" in fighting to allow the movement of aid.
Humanitarian pauses tend to last for shorter periods of time than formal ceasefires, sometimes just a few hours.
They are typically implemented purely with the aim of providing humanitarian support, as opposed to achieving long-term political solutions, according to the United Nations.