It seems to make intuitive sense for cloud movement to be dependent on wind, but is it actually true? Isn't the Coriolis Force a bigger factor in cloud movement?
What else is a cloud than a visible portion of the air? When air moves it is called 'wind'. So what if a cloud moves? Of course all motions in a rotating system are subject to coriolis force.
Note also that wind at higher altitudes usually has another direction than the surface wind. See Buys Ballots law.
For completeness, the water droplets forming the clouds are falling very slowly, independent of the wind.