I'm using Development Kit, and the character body follows moving platforms only on the vertical axis. ;(
Been giving this some more thought. With Devkit obviously Godot built in platform logic won't work seeing we use a raycast to adjust the vertical placement of the player, so the capsule never touches the ground.
If the ability to do things like climb stairs doesn't matter, you could do away with the raycast body node, and just add your own full height capsule collision shape.
Else like we talked about on stream, we'll need to enhance the logic to detect what surface we're standing on, get the velocity of the surface, and take that into account with applying friction to the body movement.