There are a lot of Child Of constraints within the gun rig. It would be better to make the subordinate bones be actual children of their main bone. Most of the gun bones would be children of the ‘Frame’ bone. The ‘Safety’ bone would be a child of ‘Slide’.
That would keep things moving together without the need for the ‘Child Of’ constraints.
The hands can use the ‘Child Of’ or an ‘Armature’ constraint to get them to follow the gun Frame bone.
The bone parenting is done in edit mode on the armature. Child Of can be a bit tricky to use correctly. There’s a risk of double-translations as the armature is moved around.
I sent an edited version of your file back via direct message. That has the parenting edited. Hopefully that will be a base to work from.