I suspect the daily thermal expansion and contraction of the rocks may cause them to creep a small increment everyday, like an inchworm. This would work only on rocks which happen to have ratchet-like grooves on the bottom in roughly the same direction, so the edges slide in one direction, but grab traction in the other direction. This would explain why different rocks move in different directions, but some don't move at all. And as the bottom surface deteriorates, that would explain why some rocks suddenly change direction.
Winter probably has more effect, with greater temperature extremes, plus ice and rain. Also the ice sheets may help steer the rocks to move more or less in the same directions. The angle of the sun would cause the ice to melt on the sunny side, but remain frozen on the shaded side, causing differential friction. Weather patterns may also influence different rocks to move and change directions in parallel. For example, if there is cloud cover every morning for a month, the sun wouldn't break till it's in the south, while otherwise it would break in the east, changing which side melts first over time.g
I imagine you could demonstrate this motion by putting a heavy metal plate with ratcheted grooves on an even wooden(?) surface in a sunny location -- most likely an indoor terrarium to rule out wind and human interference.