Consider the experiment of Gay-Lussac: A cylinder is split into two compartments $A$ and $B$ by a barrier. Compartment $A$ initially contains some gas and compartment $B$ is initially empty. The walls of the cylinder as well as the barrier are assumed to be adiabatic.
Experiment shows that if a small opening is made in the barrier, allowing the gas to pass between the two compartments, then after some time the gas will fill the whole volume of the cylinder. Indeed it's a well-known property of gases that they fill the entirety of the volume of the container in which they are confined.
During this transformation, no heat is added to the gas, and no work is done on it either. Also the net external force on the gas is zero. Initially, the center of mass of the gas is at rest, located at some point in the interior of compartment $A$. But since the gas fills the whole volume of the cylinder at equilibrium, the center of mass must move.
But if the net external force on the gas is zero, the center of mass should always remain in the same place, no?