Animating a square on a path, while also copying the form of the path

Hello Blender’s

i’m trying to create a 2D animation using blender, because it can use a “Follow Path” to make a piece of paper (cube) to direct it. i’m trying to create this:

My problem: i can animate it to follow a path, but it won’t bend to the path…
So i tried the Curve modifier, wich will make it “Curve” to the path. Problem is: it wont follow the path.

This is how far i got. Using the Curve-trick. Problem is, now it needs to “fold”. And i can’t seem to get that right.

Perhaps use an armiture to control the folding, and follow path to move everything