Why does dust seem to move along a beam
of sunlight in the same direction as the light?

I have watched dust particles in a beam of sunlight in the way you suggest. Sometimes it does seem as if dust particles are moving in the direction of the light beam. However, I think that is likely to be just an accident, or maybe our eyes fool us.

Light does exert a very small pressure, but this is so small that there is no simple way to show it. Particles in the
air are moved about mainly by small air currents. The push of light on a dust particle is much smaller than the push
of even very small air movements.