Jeppe Huldahl of Denmark (68) and Ignacio Garrido of Spain (71) are even with Dougherty at 5-under 208.
Paul Waring (70) and Simon Dyson (70) were one shot back.
After play began following heavy overnight rain and a downpour in the morning, all but one group finished their rounds. That was second-round leaders Richie Ramsey, Robert Rock and Ross Fisher.
Ramsey was locked in discussions with European Tour chief referee John Paramor afterward over a possible rule infraction at the eighth hole, but no decision was reached and the matter will be resolved Sunday.
Dougherty reached the elevated green over water at the 18th on the 2010 Course and made a 14-foot eagle putt.
I really wanted that to get to 5 under, Dougherty said. It was a tough day. The course is playing ridiculously long and there is no run on the ball. I didnt hit many bad shots but I paid the penalty when I was a little off.
The 27-year-old Huldahl, who made only one bogey, said he likes tough conditions.
I managed to stay out of trouble pretty much all the way through and got up-and-down when it was important, Huldahl said.
One of seven players at 210 was Paul McGinley (71), who won in 2001 in a playoff after only 36 holes were possible because of wet weather. McGinley, a member of three victorious European Ryder Cup teams, had four birdies and four bogeys.
I battled well but I need more quality, McGinley said. I didnt play enough quality shots to get close to the pins. I had three three-putts from 60 or 70 feet.
European Ryder Cup captain Colin Montgomerie hit a 78 for a share of 46th place. His U.S. counterpart, Corey Pavin, had a 71 to be 10 positions higher than Montgomerie.