Yes. Sort of. There are many layers of clouds and wind. The wind at the surface is (almost) always slower than the winds aloft. There may be dozens of layers of wind, in all 360 degrees of direction. "wind sheer" is the term describing a sudden change in wind direction, either within a layer, or between layers.
Don't think of the clouds as being "pushed by the wind". Masses of air (measured in cubic miles, for example) move at a speed and direction governed by pressure gradients at different locations. Clouds are part of that air mass. "Wind" is the phenomena of an air mass moving past a stationary object. So, clouds do not "move"... the air mass that contains the cloud moves.
That said, this applies to most fluffy cumulous clouds that most people experience day to day. There are all kinds of clouds that form due to interactions with other phenomina... lenticular clouds, for example, only form over mountains due to mechanical turbulence of any air mass passing over it. Those clouds are fairly stationary, as it is the rapid change in altitude the air mass is forced to take over the mountain that causes it to form.