The grid snap happens after the movement. You are correct that the higher level grid may not move. To fix this, the higher resolution tiles (1x1 in your example) are morphed near their edges so that gaps do not appear. I cover this in the blog post, see the 'Morphing between regions' section

I still don't think I understand. If the 1x1 level moves, and didn't overlap the 2x2 outer ring, and the 2x2 outer ring doesn't move (the camera hasn't triggered a 2x2 move yet, just the 1x1), then there is now a gap in the opposite direction of the 1x1 move. That's why trigger rally uses the overlapping geometries, so that a 1x1 level can move and in the direction of the move the overlapping geometries double up and in the opposite direction, they no longer overlap at all (it uses morphing also so that the overlapping geometries have vertices in the exact same spots).

Imagine instead that each ring is joined by a set of springs, which can be compressed down to zero and extended to the difference in the grid snapping between the 2 rings. As the rings snap to different grids the springs extend and contract to fill the gaps. Now place some vertices on these springs and you have a terrain mesh, where the majority of vertices are snapped to locations depending on the LOD, except those joining the rings, which such that the transitions between the layers are seamless

You are a genius :)