MacOS Mission Control strange behavior on Sierra

What you can do is this:

Delete the desktops and then recreate the by hovering over them and pressing the red thing in the corner, then looking at the right to make a new desktop.

This is no longer necessary as of Mac OS Lion 10.7.2, see this answer for an easier solution.

If the item you're trying to move springs back then it's most likely already on the space / desktop you're trying to move it to.

You can also try moving something to another desktop in ways other than Mission Control:

• Let's say you already setup two spaces, and you want something from space one onto space two; Just drag the window of whatever it is to the edge of the screen and hold it there until it moves over to the other space.
• Another way you can do this is to simply open whatever application or file you want in that particular space.

Apps are automatically assigned to the Desktop / Space where they are launched for the very first time. You change this by Control clicking the apps icon in the dock, and assign to None.