I too have various rugs which travel rapidly on a cut-pile carpet, eventually pushing up against the skirting board.
After a period of careful observation I noticed that the rug always travels in the direction that the pile leans in, and only when someone walks on it. When the foot is removed, the flattened pile returns to its rest position but, like a ratchet, does not move the rug back to its starting point.
The movement is noticeable on cut-pile carpets, but hardly at all on loop-pile carpets, suggesting the effect depends on the pile structure, the length and stiffness of the fibres, the pressure of the footstep and perhaps the structure of the underside of the rug. Incidentally, it is easier to drag a rug - with or without chair on top - across a carpet in the direction of the pile rather than against it. I have not calculated the forces involved but they must be fairly large to move an armchair.
I feel an Ig Nobel award approaching.
Donald Brown, Kellas, Angus, UK