If you just want the character to stay on the platform, you simply have to move it using physics. You can do this by unanchoring the platform and tweening an AlignOrientation and an AlignPosition.
If you actually want the character to follow the moving platform, even in the air, you’ll need to program something custom or use the character controllers below:
You can try out the new beta character controller, which has this feature:
You can also use one of EgoMoose’s controllers without the gravity modifier, though I forget which supports this: