Thomas De Gendt, Michael Schar, Dimitri Claeys and Tom Scully made up a small breakaway group of four riders and established a lead of around two minutes.
If the situation is getting to the Welshman, he was not about to show it.
"It's certainly not going to calm people down", he said. "But at the end of the day, you need to stay strong in your head and just crack on".
When Dumoulin attacked with two kilometres remaining, Thomas chased him down with Froome towing along.
Alaphilippe crossed the line second, just in front of Stuyven.
"You would have to ask the public, and maybe it's a reflection of the way we're perceived in the French media".
He was cleared of an offence days before the start of this year's race after he was found to have excessive levels of asthma drug salbutamol during one stage at last year's Vuelta a Espana.
Amid a general feeling of suspicion surrounding Sky and their sheer domination of the race, Froome has been spat at, Thomas has been booed off the podium and some of Sky's staff have also faced abuse during the opening 15 stages. "A lot of guys are speaking out about it now, the riders are sick and exhausted of it".
"It's an unbelievable position for us", Froome said.
Thomas has said that although he would work to support Froome, he is not going to intentionally lose time. "I think he is just riding so well". "I wouldn't give it up for any money".
He added: "For me it's the highlight of my career, and it's a massive honour and privilege to be wearing the jersey".
Thomas, Froome and Dumoulin all reacted to Roglic's move inside the final five kilometres, but it was too much for Bardet and Nairo Quintana who lost more time in the GC fight. "We'll take each day as it comes". Astana's Magnus Cort Neilsen won a three-up sprint with Ion Izaguirre (Bahrain-Merida) and Bauke Mollema (Trek-Segafredo) for the stage win, with the main bunch rolling home 13 minutes later.
Spanish rider Omar Fraile won the stage in the Massif Central by bursting from a large group of breakaway riders on the finishing climb, a short but steep three-kilometer ascent that was followed by a quick descent and flat finish on an air strip.
Bora-Hansgrohe's Sagan picked up his third stage win of this Tour as he overhauled UAE Team Emirates' Alexander Kristoff and Arnaud Demare of Groupama-FDJ in the final few metres to win by a wheel length.
Team Sky manager Dave Brailsford and sports director Nicolas Portal visited the race jury after the stage finish in Carcassonne to review the television footage and plead Moscon's case, however this came to no avail, with the rider being ejected yesterday evening.