Depending upon the specific arrangement, not all parts of a pulley system move at the same rate. For example: When a pulley system gives you a mechanical advantage, the applied force must travel a greater distance than the load. If that applied force is given by a hanging box, its acceleration will be greater than the acceleration of the load. (You can figure this out by studying the configuration of the ropes.)
As an example, conside a block sliding along a plane, a taut rope fixed to it going about a fixed pulley, then downwards and around a movable pulley, the rope then going upwards to be fixed in the ceiling.
In this case, inextensibility of rope requires that the acceleration of the movable pulley is only one-half of the acceleration of the block.