OSX Lion 10.7.1 - Safari, Textedit or Stickies in separate Mission Control Desktops

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.