The way I made jump-through platforms in my game is if the bottom edge of the player Y is greater than the top edge of the platform Y, then the platform is not solid. If the beopY it less than the teopY then the platform is solid. This worked fairly well unless an enemy happens to be passing through the platform at the same time the player is.
All of my experiments with making jump-through platforms have worked to a certain extent, but they always seem to have small problems like that. For instance, if you make a custom push-up routine where the platform automatically updates the player to the top of the platform if they're falling, then it's possible to miss the top edge of a platform, and enter it from the side and get pushed up anyway, which just looks funny. I suppose you could get around this by making your platforms 1px tall, but that would be a pain to work with in the layout editor.
The Platform object for MMF handled this kind of thing automatically, where you could specify "player is overlapping a jump-through platform" for certain objects. It worked great. In that regard, the Platform object was one up on Construct's.
I was thinking of requesting that there be more than one kind of "Solid" attribute, just for this kind of thing... objects that are only solid if passing through on one side. Like "Solid on top," "Solid on left," etc. I don't know how useful that would be to a lot of people though (It would be useful to me for sure).
As for the moving platforms, it's a matter of adding the speed of the object to the speed of the player. There are a few different ways you could implement that, but I'm sure you can come up with something. In my experiments, making horizontal platforms is way easier than vertical ones. My vertical ones always make the player jitter around, and I can't seem to get the motion smooth.
Again this is where the MMF Platform object has a leg up on Construct... there were two different speed settings for the PMO: Velocity and Additional Velocity. If a platform had Velocity x and the player was on top of it, all you had to do was say "Set Additional Velocity to platform velocity x" and it would automagically work. So if the player was running at 50, and the platform is moving at 50, then the total speed of the player would be 100. Easy. It's a little more complicated in Construct, but it is doable.
I'll mess around with an example and post it later, but right now I have to watch the season premier of Bones.