After a dye diffuses uniformly throughout a glass of water, the dye molecules are no longer moving. True or false?

The answer is false, liquid molecules continue to move even after the concentration is uniform.

Diffusion happens because there's a concentration gradient between the dye and the water, therefore dye molecules moves to the water area. But note that this is only the net movement of the molecules, which means, even if more molecules are moving towards water, there are still some dye molecules that are moving in the opposite direction.

So, after the concentration is equal, the motion of the dye molecules does not stop, they still move in random directions, but there's no net movement instead, as there's no concentration gradient. Remember molecules are always moving in all directions, it just depends on whether there's a net movement or not. If there's no net movement, the amount of molecules moving to each and every direction is always equal.