The Swift Brigade is the regiment of the Royal Army stationed in Mourningwood Fort and commanded by Major Swift. It is the unofficial name of the regiment, and reflects the soldiers' loyalty to the eponymous Major.
Major Swift reminisces with Walter about fighting alongside the Hero of Bowerstone when he/she was uniting Albion, though it is unknown if he was leading the Swift Brigade at the time. If so, it would indicate that they were among the Albion Royal Army's first regiments.
During King Logan's reign, Major Swift and his soldiers somehow proved themselves as less-than loyal to Albion's dictator. In response, Logan intentionally sent Major Swift and his soldiers to defend Mourningwood Fort against the seemingly-infinite legions of hollow men, hoping to silence them by stationing them in such a remote area and possibly getting killed by the aforementioned hollow men. It is then that the brigade is saved from the forces of the Hollow Legion by the Hero of Brightwall, who not only saves the soldiers, but destroys the enroaching hordes of hollow men. Impressed, Major Swift and the Swift Brigade throw in their lot with the revolution, desiring a chance to help rather than oppress the people of Albion. They prove themselves to be fiercely loyal and hard fighters, helping win the Battle for Albion and installing the Hero as monarch.