In the 21 months since the referendum, May, who became prime minister in the resulting political chaos, has struggled to unite the country behind a single vision of Brexit.
The Irish border remains a key difficulty in Brexit with no solutions yet as to how it might be addressed.
Shadow chancellor John McDonnell defended Ms Thornberry's comments, emphasising her "sarcasm" in response to a journalist's question.
Meanwhile, speaking to the Independent, ex-PM Tony Blair has urged MPs to vote "according to what they genuinely believe", even if it means defying their party.
On Thursday, Mrs May will visit a textile factory in Ayrshire, a parent and toddler group in Newcastle, have lunch with farmers near Belfast before meeting businesses in Barry, south Wales.
"As we leave the European Union, powers will return from Brussels to the parliaments and assemblies of the United Kingdom, closer to the people we all serve and with greater ability to deliver for their needs".
Scottish National Party spokesman Michael Russell accused Mrs May of planning a "power grab" by repatriating some responsibilities now exercised in Brussels to Westminster rather than Edinburgh.
Mrs May insisted that each of the devolved administrations will see "an increase in their decision-making powers" as a result of the return of responsibilities now exercised by the EU.
"It's very roundabout and complicated but essentially yes there is a dividend from leaving the European Union".
The British prime minister kicked off a whistle-stop tour of the United Kingdom with a visit to a weaving firm in Ayr, where she said she expected to be able to negotiate a deal with the European Union allowing a "good trading relationship" in future. People did feel that the system wasn't working for them in perhaps the way that it should.
Mike Hawes, SMMT chief executive, said: "Another month of double-digit decline in production for the United Kingdom is of considerable concern, but we hope that the degree of certainty provided by last week's Brexit transition agreement will help stimulate business and consumer confidence over the coming months".
"There's a whole list of something like 150 powers, the vast majority of them are actually powers that are going to be directly devolved to the Scottish Government".
Mrs May also set out a vision of economic prosperity at home, and influence overseas, trading freely with friends and partners across Europe and beyond.
May also told the BBC that Brexit "would deliver a country that will be different" but with the chance "of a bright future".
