Head for Caliga Hall and address the guards before heading into the stables to meet up with John and Javier. After you learn about the horses, ride with the gang until you reach the Braithwaite stables. Take the lead with Arthur and when given the opportunity, perform a Stealth Kill on the stable hand. Head towards the white horse in the center and calm it so that you can lead it outside to John.
The escape sequence will begin now, continue alongside the other gang members and make use of Dead Eye to headshot at least 5 of your pursuers. You don’t need to worry about accuracy or using items for the Gold Medal here so you don’t need to be too precise as long as you nail the headshots. After the last of the guards fall, you’ll notice the right horse begin to buck a little, equip the Lasso and get ready. Once the horse breaks free, use the Lasso immediately to catch it, you only have 10 seconds to achieve this for the Gold Medal. Be sure to keep the [L2]/[LT] button held in as you lead it to the marker and complete the mission.