Can you help me to manage reloading animations with two rigs?

So I’m animating a character with a gun. The gun has one rig and the character in question has another rig. I managed to control the movement of the gun with the character rig by using the the child of constraint. However, I have this issue where both the rig for the character and the gun must be linked to the same animation at the same time in order for the animation to play for that one animation to play for both rigs. If only the character rig is linked to the animation, then only their part of the animation will play while the part of the animation for the gun will not and visa versa. Now this might not seem like a big issue but, handling these animations is getting tedious; it would be nice if linking the animation for one rig would automatically like the same animation for the other rig or sometime like that. Any help?