The moving platform tends to lag, causing players to occasionally fall through it. It is a better idea to remain on the left side and wait for a bridge to be built.
Take turns riding the platform over to the cheese, and back to the mouse hole. Alternatively, wait until the Shaman builds a bridge that bypasses the platform. A good way to avoid lag is to jump to the side of the platform facing your destination while other mice push. That way, if it lags, you will merely go to the other side of the platform instead of being thrown over the edge.
Build a bridge from the starting platform over to the cheese, completely bridging the gap. Ensure no mice are trapped on the moving platform. Building small B anchored planks under the bridge is a good way to hold it.