A series of senior Conservatives have contradicted Kemi Badenoch after she criticised a landmark UK-India trade deal because it temporarily exempts seconded Indian workers from national insurance payments in the UK.

Tories including Oliver Dowden, who was deputy prime minister under Rishi Sunak, said the deal should be hailed as a dividend of Brexit that would bring economic growth and cheaper goods from India.

The deal was announced on Tuesday after more than three years of negotiations. It cuts tariffs on a series of goods and will add an estimated £4.8bn a year to the UK economy by 2040.

In an initial response, the shadow trade secretary, Andrew Griffith, praised it, saying it showed the government recognised “that reducing cost and burdens on businesses in international trade is a good thing, and that thanks to Brexit, we can do”.

But later on Tuesday the tone changed, with Robert Jenrick, the shadow justice secretary – who regularly roams beyond his brief – tweeting that the national insurance exemption, which applies mutually to seconded UK workers in India, showed that “British workers come last in Starmer’s Britain”.

Badenoch, the party leader, soon followed suit, saying in a tweet that this was “two-tier taxes from two-tier Keir”.

But several influential Tories and figures from the pro-Brexit camp pointedly disagreed, noting that such opt-outs for seconded workers, which prevent double taxation, were routine in trade deals and had featured in some negotiated under the Conservatives.

Dowden, who is still an MP, welcomed the deal, writing on X that it “builds on significant progress made by [the] previous Conservative government”.

Steve Baker, who dealt with trade as a Brexit minister under Theresa May, wrote: “This deal is great news. It further cements the path which I and others worked so hard to secure … The tax issue will likely turn out to be a red herring. We should be celebrating that a Labour government has furthered free trade in the national interest outside the EU.”

Another leading Tory Brexiter, Jacob Rees-Mogg, who was business secretary under Liz Truss, tweeted: “Cheaper food and drink including rice and tea, footwear and clothing thanks to a welcome trade deal with India. Exactly what Brexit promised.”

Praise for the deal – and scepticism about Badenoch’s view – also came from some influential Brexit campaigners. In an opinion piece for the Telegraph, Daniel Hannan, a Tory former MEP who is now a peer, wrote that the UK had “pulled off something that no other country has, at least not on anything like the same scale”.

Noting that some people had criticised the deal based on the tax issue and worries about its impact on migration and apparently uneven tariff reduction, he wrote: “All three are nonsense.”

Shanker Singham, a pro-Brexit trade economist who advised Liam Fox when he was international trade secretary, wrote on X: “This is a significant achievement for UK trade policy. If the UK can lock in a deal with the US, it will be one of the few countries with deals with the key trade players.”

Asked about why Badenoch objected to the deal, her spokesperson said the worries were less about the national insurance idea in principle than the fact the deal was with India, given the size of its population and its lower wages.

“The fear is that because Indian workers are paid considerably less, because their social security contributions would be considerably less, there is an obvious route for companies to bring Indian workers over at the expense of British workers,” he said.

Asked how this would happen given the exemption only applied to workers who were temporarily seconded within the same company, the spokesman said there were “ways of setting up companies in countries, and then bringing workers over”.