Internal Forces In a System

My high school teacher told me that internal forces of a system do not change the Centre of Gravity of a system. I have doubt regarding this.
If internal force is applied to the wall of a system, the CG must change right? If not, where'll be the result of applied force.
I have some examples that I think are related to this:

1) Consider zorbing. Here, the ball rolls if you apply a force from within. So, that counts as an internal force.

2) Consider a child swinging in a park. Once we push the swing, it keeps swinging until it stops. Now, if the child stretches its legs and folds them during forward and backward motion, the total time of swinging is prolonged. Since there is no external force, the child's movement can be considered as an internal force. So, internal force prolongs motion.

[Fiction Ahead]
3)Finally, if you have seen Pirates of Caribbean At World's End, you'll know what Jack Sparrow does to get out of the other world. There, the barrels hitting inside the hull of the ship inverts it. They are internal forces too. (I know this is not totally appropriate but it came to my mind and I posted it)

Someone please resolve my doubt.

Prajwal

EDIT: The teacher said the position of CG is not changed.