At the end of Chapter 42, Moiraine leaves Lan (and the rest) very explicit instructions to stay behind while she scouts ahead. Lan, Perrin, Faile, and Loial take time to scout out the immediate surroundings of their inn. Discovering paw prints from a Darkhound etched clearly in the cobblestone streets, Lan elects to disobey Moiraine.
The Warder was silent for a time. "The answer to that, blacksmith," he said grimly at last, "may be more than you or I, either one, want to know. I hope the answer does not kill us all. You three get what sleep you can. I doubt we will stay the night in Illian, and I fear we have hard riding ahead."
"What are you going to do?" Perrin asked.
"I am going after Moiraine. To tell her about the Darkhound. She can't be angry with me for following for that, not when she would not know it was thereuntil it took her throat."
Lan and Perrin, The Dragon Reborn, p. 424
Actively disobeying a direct order is a bold choice. For Lan, it's a matter between his original oath to protect Moiraine and her more recent demand that he stay behind to protect Perrin, Faile and Loial in the event of her death. Lan's decision rests entirely on his interpretation of Moiraine's intent. The Darkhound is a new variable to the equation, one that Moiraine did not foresee or plan for. This new information demands immediate action. With no other means of communicating the hazards to Moiraine, Lan follows her; leaving the other three behind. No doubt he feels that Perrin's ability to smell the evil of the Gray Men, and others, leaves them in as good a defensive position as any at the moment. Lan's success is in acting on his oath to defend Moiraine. She might be angry, but at least she'll be alive to be angry.
In the end, it comes down to trusting folks to make decisions and act quickly and advantageously in the leader's absence. Leaders facilitate this by communicating intent, continually driving decisions down to the lowest effective level and keeping an open mind when taking the reports upon returning to the team.