Okay, no solution for the vertical movement, yet, but I think I know the root of the problem that also leads to my Player not sticking to the moving platform.
My Platform Moving behaviour uses the "Slide to/by" block which doesn't seem to use a speed value but rather moves the actor manually step by step, I suppose. I debugged and it looks like the platform's speed is 0.
Hmm, I guess I have to rewrite the platform motion code :/