First responders have a black band that is worn over or around their badge/shield like what is pictured. The band signifies the loss of a colleague usually through a Line of Duty Death (LODD); the military equivalent to this would be an Active Duty Death.
Three questions arise:
How long should the department wear the band?
Thirty days is the usual standard for a department in mourning.
If the honor guard has ceremonies to perform within that 30 days that are not involved with the LODD, is it appropriate to still wear the band?
Yes, it is quite appropriate. The team is in mourning along with the rest of the department. The honor guard may deem it necessary to create guidance that requires the team to remove the band when performing other ceremonial duties (e.g. a parade).
Should the honor guard wear the band all of the time?
That would lessen the meaning of why the band is worn. While the team exists to provide honors for fallen colleagues, that is not its sole purpose.
Featured image courtesy of the Civil Service Supply Company.