Promises to quickly reduce immigration levels are also being played down. Migration was the cornerstone of the Leave campaign, which objected to the European Union’s insistence on the free movement of labor, capital, goods and services. Since 2004, when 10 more countries joined the European Union, large numbers of eastern and southern Europeans have moved to Britain for work.

Mr. Johnson argued that it was impossible for the government to reduce immigration while in the European Union. His ally Michael Gove, the justice secretary, said a leave vote would “bring down the numbers” by 2020. Experts have long said that would be very hard to pull off. The European Union has demanded from nonmember states — Norway, for example — free movement of workers in exchange for access to the bloc’s single market.

On Friday, the day after the referendum, Daniel Hannan, a member of the European Parliament and one of the most knowledgeable advocates of Brexit, stunned some viewers of the BBC by saying: “Frankly, if people watching think that they have voted and there is now going to be zero immigration from the E.U., they are going to be disappointed.”

Mr. Hannan wrote on Twitter, “I was for more control, not for minimal immigration.” Facing a backlash, he observed that a lot of Remain voters “are now raging at me because I *don’t* want to cut immigration sharply,” adding, “There really is no pleasing some people.” He then announced that he would “take a month off Twitter.”

Mr. Farage took the most hard-line position on immigration, unveiling a poster depicting a long stream of refugees under the headline “Breaking Point” and raising the prospect of Turkey’s joining the European Union (even though any of the 28 member states has veto power over accepting new members).

Many Brexit campaigners expected to lose — even Mr. Farage said on the night of the referendum that he did not think his side had won — and for some the fight was as much about internal Conservative Party politics as the future of the country.

Having now ousted Prime Minister David Cameron, they face a political vacuum, with their base demanding that promises be kept. Mr. Johnson, the front-runner to replace Mr. Cameron, has not made any further pronouncements since a subdued statement on Friday that was restricted to generalities. If he does become prime minister, Mr. Johnson will face the task of carrying out a British withdrawal without provoking a backlash from those who believed campaign slogans or sentiments that he certainly appeared to endorse.

“In voting to leave the E.U., it is vital to stress that there is now no need for haste,” Mr. Johnson told reporters. Standing alongside him, Mr. Gove promised, “We can have democratic consent for an immigration policy that is fairer and more humane.” He did not explain how.