When you select a mission from the Region screen, you will enter the Loadout, which will display the currently equipped weapon that matches the type required by the mission. For example, when attempting to start a Trophy Hunt, you will always be presented with your currently equipped rifle. You may switch this weapon to another of the same type by pressing the Switch Weapon button on the navigation bar at the bottom left corner of the screen. You will only be able to swap to a weapon that you own of the same type.
While the Loadout screen only presents weapons of a limited type to you, while in the mission you may press the Swap button on the bottom right section of the screen. Pressing this button will pause the game and bring up a menu that allows you to select from any of your currently equipped weapons based on weapon type. This will allow you to switch between your Rifle, Shotgun, and AR.