The home secretary, Sajid Javid, and his French counterpart have pledged to step up joint efforts to tackle cross-Channel people smuggling, as Labour accused the Conservatives of whipping up concern about the issue.

On Monday, a group of 12 people, including a 10-year-old child, were being interviewed by immigration officials after they came ashore near Lydd-on-Sea, Kent.

A Home Office spokesman said: “Border Force was contacted by Kent police at around 8.25am. Border Force officers were deployed to assist with a group of 12 migrants at Greatstone, Kent.

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“The group consisted of nine men, two women and a 10-year-old child. They have presented themselves as Iranian nationals. They all received a medical assessment and have now been transferred to immigration officials for interview.”

The latest incident came after six Iranian men were found on a beach near Deal in Kent on Sunday morning with a small boat, bringing to almost 100 the number who have made the perilous crossing since Christmas Day.

Javid flew back early from a family holiday in South Africa to take direct control of the situation, which the government declared a major incident on Friday, although Home Office sources said there was no clear evidence of a sudden sharp increase in the number of arrivals via the Channel.

Javid and the French interior minister, Christophe Castaner, are expected to launch a joint action plan later this week. It will include further patrols in the area, enhanced intelligence sharing aimed at breaking up the gangs involved, and an information campaign to raise awareness about the risks of the crossing.

The pair had what the Home Office described as a significant and productive phone call on Sunday, after which Castaner said they had agreed to “strengthen our actions to combat Channel crossings undertaken by certain irregular migrants on small boats, at peril of their lives”.

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He told Javid that French authorities had dismantled a people-smuggling gang operating in the area on 19 December.

In an article written for the Daily Telegraph, Javid admitted “the reasons behind the increased crossings are complicated and in many cases outside of our control”. He cited factors including instability in the Middle East, organised crime and tighter security at Calais, adding: “Unfortunately there are no easy answers.”

Javid will chair a meeting on Monday that will bring together senior officials from the Border Force, the National Crime Agency and other authorities.

The shadow home secretary, Diane Abbott, said it was right to increase cooperation with the French authorities, because vulnerable migrants were at risk in one of the world’s busiest shipping lanes.

She also accused the government of exploiting the issue. “There’s no question that with Brexit, and also with the approach of the meaningful vote in January, people are being whipped up about migration issues, because the government thinks this is the best way of frightening people to vote for their deal.

“It’s the Farage technique: ‘hordes’ of people trying to enter the country. You frighten people about that.”

Abbott said Brexit, and particularly a no-deal Brexit, would make it harder to cooperate on tackling the underlying issues. “We need to work much more closely with the authorities on the ground in northern France. But of course Brexit makes that harder, not easier. We’re going towards Brexit without a security treaty, and the possibility of working with the local people on the ground about this are fast diminishing.”

The official Vote Leave campaign used the risk of a sharp increase in legal migration, including from Turkey if it ever joined the EU, as an argument during the 2016 referendum. Nigel Farage’s Leave.EU also pointed to the unrelated increase in refugees and asylum seekers as a reason to quit the bloc.

At the weekend, the defence secretary, Gavin Williamson, offered to deploy the navy to deal with the small boats trying to cross the Channel. He told the Sunday Telegraph: “We have not had any requests as yet, but if the Home Office is in need of armed forces support then our navy, air force and army stand ready to assist.”

A government source said: “I wonder if he realises that if they strayed into French waters it would be an act of war?”

Abbott said: “I think Gavin Williamson is just nakedly trailing his coat for the leadership, and I think that’s a little bit unpleasant, given that these are real human lives at stake.”

The MP for Dover, Charlie Elphicke, said he welcomed the home secretary taking personal responsibility for the situation. “Illegal crossings have now been on the increase for more than two months and more needs to be done as a matter of urgency,” he said. “This is a good start. It is right that our government has been demanding more of the French authorities to prevent further crossings, which put the lives of innocent and desperate people at risk and fuel criminal gangs.”

He repeated calls for Border Force cutters to be called back from search-and-rescue operations in the Mediterranean to help patrol the Channel.

Lucy Moreton, a spokeswoman for the Immigration Services Union (ISU), which represents Border Force staff, said the boats available for patrols in the channel were “woefully inadequate”. She also said it was very difficult to know how much the French authorities were doing to prevent people smuggling.

“We are being told that those touting for these crossings are absolutely open about it,” she told BBC Radio 4’s Today programme. “They are around and about in the camps, they are in the cafes in those areas of Calais.”

The immigration minister, Caroline Nokes, has argued that deploying more vessels to the area could exert a pull factor over potential migrants. “They might act as a magnet, encouraging people to make a perilous crossing,” she said.

Abbott dismissed that argument. “Any type of immigration policy that you enforce by leaving people to die is the wrong immigration policy. We had the same arguments about people crossing the Med. If you save them, it’s a pull factor.

“Actually, the push factor is how desperate these people are. If you even begin to justify allowing people to drown in the sea by talking about pull factors, it’s quite immoral,” she said.