Riding Shotgun boosts gun damage from carriages after fifty kills shot from a moving carriage roof. Carriage chases are where it comes up.
Riding Shotgun unlocks after this challenge: kill an enemy with your gun from the top of a moving carriage, 50 times. As with the other 28 base perks, the Riding Shotgun challenge also counts when you play Lydia in the World War I simulation.