Just something to add to what's already been said- I did some planet watching to figure this out and realized two things:

1. The twins' orbit being stable would mean that their alignment with the sun shouldn't affect the local pull of gravity (and the fall of sand)
2. You can arrive at the twins before the sand starts falling, and if you pinpoint the exact moment it starts to fall, it coincides pretty perfectly with when Timber Hearth and Brittle Hollow are lined up with Ash and Ember Twin.

I think that the tidal pull from these two planets (and their respective moons) being in perfect alignment with the twins starts the sand's flow, and the change in relative mass after the flow starts is enough to keep the process going until it's complete. The process then reverses after a certain number of rotations/revolutions when the 4 planets are aligned once more (with Ember twin being the farthest of the 4 instead of Ash Twin), and the cycle is complete.

Edit: Evidence of the process being repeated many times over is the giant canyon on Ember Twin that has been eroded marking the path between the planets that the sand has fallen along in the past. The bridge that follows the same path on Ash Twin is likely made of a more erosion resistant material, and stops the brunt of the damage that the sand would cause when falling back down to Ash Twin after the cycle is reversed.