
Considering the overuse of Adventurers in Eorzea and the degree to which they operate in the affairs of city-states, it can be assumed that anyone with sufficient investment can apprehend a criminal to be turned over to the Brass Blades. Immortal Flames and Sultansworn acting outside of their specific duties wouldn't be empowered to act anymore than any other citizen.
However, no one likes their toes being stepped on, especially an organization with as much corruption as the Brass Blades (a fact that doesn't invalidate their authority). If a member of the Flames, the Sultansworn, or an Adventurer apprehended someone without a bounty, they may be subject to punishment for overstepping their bounds. Actually killing a criminal would get the offender arrested by the Blades, regardless of what other position they hold, pending a trial (or bribe) to prove that they acted in the interest of the city (or the Syndicate's agenda).
Ul'dah isn't a lawless city where people are murdered in the streets daily. There are heralds that announce murders and the Brass Blades' investigations into the crimes. They don't like other people doing their job, and they have the power to act against the people that try. The Immortal Flames and the Sultansworn have their respective duties, and those don't include murdering suspected criminals in the streets.
However, no one likes their toes being stepped on, especially an organization with as much corruption as the Brass Blades (a fact that doesn't invalidate their authority). If a member of the Flames, the Sultansworn, or an Adventurer apprehended someone without a bounty, they may be subject to punishment for overstepping their bounds. Actually killing a criminal would get the offender arrested by the Blades, regardless of what other position they hold, pending a trial (or bribe) to prove that they acted in the interest of the city (or the Syndicate's agenda).
Ul'dah isn't a lawless city where people are murdered in the streets daily. There are heralds that announce murders and the Brass Blades' investigations into the crimes. They don't like other people doing their job, and they have the power to act against the people that try. The Immortal Flames and the Sultansworn have their respective duties, and those don't include murdering suspected criminals in the streets.