That's the default behavior--after you move, the next ready move comes automatically. Now, to stay on a particular game, you have to change your Online Games settings. Go to your Online Chess, then click Settings. The Preferences - Online Chess appears. Scroll down to Move Preferences, and choose an option at the After Moving text box by clicking the drop-down arrow. Choose Stay on this Game, and then click Save Preferences button at the bottom of the page.
Now, when you choose a game (from My Games), you'll stay with it even if you submit your move.
From my list of games I choose one which is "My Move" and I wish to play. After I submit my move another one automatically comes up.
If I have another "My Move" game, is there a way to jump directly to the next game, without making a move on the one presented to me?
Sometimes, I just want to wait on one game if I have the time.