But a group representing them said the Foreign Office left those without UK passports off the safe passage list.
Officials begun putting names of UK citizens on to the list of those allowed to leave Gaza via the Rafah crossing to Egypt - the only route for foreign nationals and sole entry point for incoming aid.
The Palestinian border authority has been issuing lists of those who can present themselves at the crossing with their passports. Friday’s named more than 90 British citizens, with 88 on Saturday.
Before Mr Dowden's comments, it was not clear how many had actually managed to leave.
Support Families in Gaza said the Foreign, Commonwealth and Development Office’s policy is in “stark contrast” to the decisions made during the evacuation of Ukraine, where any family member of a British citizen would be provided with a visa.
It also differs from those set by other countries, with Britons reporting the US is allowing anybody with an American family to leave, the group claims.
British citizens were among those documented on an approved list to pass through the key border post on Saturday but many reported being turned away in an apparent row between Israel and the Palestinians over evacuating injured patients.
One person facing the difficult prospect of leaving family members behind said those with Palestinian heritage were being treated like “second-class citizens”.
The person – who did not wish to be identified – said: “When we got the call that I and other family members had been added to the list we found out that my dad’s wife is not on there.
“My dad has had to leave his wife and other family members behind so he can get my young brothers to safety.
“We know of at least one other family in this situation. It is heartbreaking, nobody should have to make this kind of choice. The world has lost its humanity.
“It is even more difficult to understand when we know of somebody who lives in the US who has managed to get 19 family members evacuated even though not even one of them has an American passport.
“We know FCDO have made exemptions in the past for people from Ukraine, Afghanistan, and Sudan, it just feels like those of us with Palestinian heritage are always treated like second-class citizens.”
Louise Harkin, of Support Families in Gaza, said: “We have been in touch with organisations and lawyers who work to support people who have a right to come to the UK through different visa schemes, they have been quite clear that British Palestinians are being treated very differently to people fleeing the Ukraine conflict.
“Gaza is currently the least safe place on the planet, almost 10,000 people are confirmed dead, almost half of whom are children, yet the Government now wants to separate British children from their mothers and families. We call upon the Government to allow British citizens to bring their families with them.”
The fighting between Israel and Hamas has continued into a fifth week, with the violence having been ignited by Hamas’s bloody raids on October 7 that killed 1,400 people and saw about 240 people taken hostage.
The Hamas-run Gaza Health Ministry has said more than 9,400 Palestinians have been killed in the enclave in nearly a month of war.
Even British nationals who are on the list have faced difficulties crossing the border in recent days due to disagreements between Palestinian and Israeli authorities.
Zaynab Wandawi, a British national born in Salford, Greater Manchester, was turned away again at the Rafah border crossing into Egypt on Saturday, her mother Lalah Ali-Faten told reporters.
Ms Wandawi, an English language teacher, and a group of 12 family members – 10 of whom are British nationals – attempted to cross the border into Egypt on Wednesday, but were told their names were not on the list.
The 29-year-old travelled to Gaza at the beginning of October with her husband, who is British Palestinian, and his relatives for a family member’s wedding before the Israel-Hamas war erupted.
Her mother said she believes the Foreign Office is “not doing enough”, and added: “Just passing the list of names, I’m sorry, that doesn’t cut it as a government.”
The UK Government was “hopeful” the Rafah crossing would reopen on Sunday to allow more British nationals to leave Gaza.
But it was confirmed the key border post remained closed to foreign nationals all day.Dowden said Britain was working to ensure civilians could exit the besieged territory “as rapidly as possible”, with Foreign Office officials looking to “facilitate them in getting to the border and crossing” over to Egypt.
It was thought there were around 200 British nationals in Gaza before war broke out.
Dowden added: “The first thing we are doing is trying to make sure we get the Rafah crossing open again and I’m hopeful we will make progress on that today.
“Secondly, we are seeking to have these temporary pauses to allow humanitarian aid in and to get our people out.”
Among Britons who have left are the in-laws of Scotland's First Minister Humza Yousaf.
Elizabeth and Maged El-Nakla - the parents of Mr Yousaf’s wife Nadia, were allowed to leave via crossing on Friday. They had been trapped there after war broke out while they were visiting their son and grandchildren.
Writing on X, formerly Twitter, Mr Yousaf said: “We are of course elated, but my father-in-law said ‘My heart is broken in two, with my mum, son and grandchildren in Gaza’. He then broke down telling me how hard it was saying goodbye to them.”
The Government said the advice remains for all British nationals and dependants to leave Gaza.
In a statement, the FCDO said: “This continues to be a complex and challenging situation and we are using all diplomatic channels to press for the crossing to reopen in co-ordination with our international partners.
“We remain in contact with British nationals in the region to provide them with the latest information.”
People who require a visa will need one in advance of travel to the UK, including dependants of British nationals, such as spouses, partners and children under 18, the FCDO said.
Israel has reportedly been trying to quietly push for civilians stuck in Gaza to be transferred to Egypt.
Six senior foreign nationals quoted by the New York Times said Israel’s diplomats have proposed the idea of moving hundreds and thousands to refugee camps in the Sinai Desert.
But this was essentially rejected over fears it would permanently displace Palestinians from the land – something the Middle East fears as a second “nakba”.
The Arabic word, which translates to catastrophe in English, is used to refer to when more than 700,000 Palestinians ended up leaving their homes in the 1948 war, when the current state of Israel was created.
While Egypt has not commented directly on reports of Israel’s supposed proposal, it has previously rejected the idea in any form.
President Abdel Fattah el-Sisi said in a speech last month: “Egypt has affirmed and reiterated its complete rejection of the forced displacement of Palestinians and their exodus to Egyptian lands in Sinai, as this is nothing but a final liquidation of the Palestinian cause.”
Israeli Prime Minister Benjamin Netanyahu’s office declined to give a statement on the New York Times article. The Evening Standard has also contacted it for comment.