Molecules never stop moving, even when equilibrium is reached. Chemical equilibrium is reached when forward and reverse reactions occur at rates that cancel each other out, and there is no net change in reactants versus products. Thermal equilibrium is reached when two touching substances reach the same temperature.
Chemical equilibrium is reached when the forward reaction occurs at the same rate as the reverse reaction in a system. This means that while there is no overall change in concentration of products and reactants, reactions are still taking place.
The same concept applies to thermal equilibrium. Molecules flow from hotter substances to cooler substances. When the two substances reach the same temperature, molecules are still transferred between the two, but at equal rates back and forth.