Do molecules stop moving when diffusion stops?
No - particles continue to move even after they are evenly spread out by the process of diffusion.
Molecules are always moving. Diffusion just means the movement of molecules from the region of higher concentration to the region of lower concentration. When a molecule is diffusing, the molecule moves and when it reaches equilibrium, it means that the movement of the molecule from the higher concentration region to lower concentration stops but the molecules are always in a random movement about their fixed position.
The continued movement of particles after diffusion spreads out particles is called dynamic equilibrium.
Here is a video which discusses these topics using a computer model.
Hope this helps!