The Frenchman - himself a town mayor - was elected International Cycling Union (UCI) president last September and had to deal with Sky rider Chris Froome's positive test for excessive levels of an asthma drug in last year's Vuelta a Espana.
The Briton was cleared just before the start of this year's Tour de France, but he has been jeered and booed since the race started on Saturday. Lappartient said the affair had damaged the image of cycling and that he would have preferred Froome to withdraw from racing while the case was being investigated.
"I gave him the benefit of the doubt when he started," Brailsford said of Lappartient on Sunday. "I thought, 'OK, he is new to the job, he obviously doesn't quite understand the responsibilities of a presidential role.' I think he has still got the local French mayor kind of mentality."
The remark struck a chord as Lappartient has been the mayor of the Brittany town of Sarzeau, where the fourth stage of the Tour ended on Tuesday, for 10 years.
"When you have his (Brailsford's) level of popularity, you'd do better to keep a low profile," Lappartient said on Tuesday.
"Who hosts the Tour de France stages? The French mayors. So it’s an insult to all the 35,000 French mayors and to French people in general."
(Reporting by Julien Pretot; Editing by Kevin Liffey)
