The lances weigh roughly 1kg and have a metal tip on the end.
Fulton said the lance's tip went into Barclay's abdomen.
"He got off the horse, took some steps and people noticed he was bleeding," Fulton said. Barclay then collapsed and was air lifted from the event to a hospital but died en route, according to Fulton.
Fulton said his group is cooperating with investigators as they look into what went wrong. He said his group follows a "very strict set of rules" and safety measures for its "combat-related activities."
Fulton said Barclay was wearing a doublet, which is a short, tight fighting jacket. He said he was not wearing full body armour, which is not necessary for the activity he was doing, given that it was just a timed event with no other riders directly in the ring at the time.
Barclay had been involved in Medieval events and reenactments for more than 30 years and taught others how to ride horses and do the activities, Fulton said. The group has more than 30,000 members, including groups in the UK, Austria, Australia and Denmark.
Barclay was known for doing the event he participated in two or three times a month and was considered a leader in the group as its deputy for equestrian activities. Fulton said Barclay had recently retired, having served the last four years at the Pentagon.
"He was the consummate expert," Fulton said. "He knew how to do it and how to do it safely."
"It was just something that happened and we still don't have a grasp of it."
While there has been injuries before, Fulton said it is the first time the group has had anyone die. He said it was "shocking to have this happen to one of the best people in our organisation."
"It is a horrible set of circumstances that caused this."