The general 'driving force' behind attainment of thermal equilibrium and dynamic chemical equilibrium When two systems are in contact with one another, energy will transfer as heat from the object at higher temperature to the one at lower temperature until they both reach the same temperature. The objects are then in thermal equilibrium, and no further temperature changes will occur if they are isolated from other systems. What drives these systems to thermal equilibrium is the increased entropy of the universe i.e the increased energy dispersal of the universe.
In a reversible chemical reaction the forward reaction must start, before the products can appear. As a result, the reverse reaction’s rate starts off at zero. However, as products begin to appear, and their concentrations build up, the rate of the reverse reaction gradually picks up speed, while the rate of the forward reaction slows. This slowness is caused by a decrease in the concentration of the reactants. As the reaction continues, it gets to a point where the rate at which the reactants react to form products is equal to the rate at which products react to form back reactants. When this point is reached, the reaction is in a state called dynamic chemical equilibrium. 
It therefore seems to me that the attainment of dynamic chemical equilibrium can be explained in terms of the increasing and decreasing rates of the forward and backward reactions of a reversible reaction. Is the underlying 'driving force' behind this the increased entropy of the universe i.e the increased energy dispersal of the universe as a result of attaining dynamic chemical equilibrium?
 A: Yes, though entropy increase is not a fundamental physical law but a consequence of probability. The fundamental assumption in Statistical Mechanics is that all accessible microscopic states are equally probable. So one defines the probability of a macroscopic state as proportional to the amount of microscopic configurations $\Omega$ that are compatible with the given macroscopic parameters. 
You can think of your reversible chemical reaction as an expanding free gas, initially confined to one half of a container. Your forward reaction are particles moving to the empty side, your reverse reaction are particles moving back to the full side. There are twice as many ways to arrange the particles so as to occupy the whole container as to only occupy one half, so the gas expands to its most probable state. 
The entropy of an isolated system like this is defined as $S=k\ln \Omega$, so it also increases (it does so by an amount $k\ln2$). There is actually a nonzero probability of all the particles going back to one side; but if you do the calculation, you will find you will have to wait much longer than the age of the current universe to see it (and furthermore would exist only for a very short time before collapsing back into the full container).
