My problem is when the player lands on the platform, as the platform moves along the X axis - the player is not moved along with it, as you visually would expect.
The player can land on the object, and if it hits the side of the object, it does colide with it and is pushed.
This image might explain better than I did:
After jumping on to the red platform the player character will fall off as the platform moves to the right
Here is a live demo showing the problem: