If you have no more games awaiting your move or if you've unchecked the move to next game waiting box, after clicking move, the game will refresh to the current game where the move was just made including a view of any message just sent. Sounds as clear as mud to me - maybe someone can explain it a little better.
Ignore my post above the same has just happened to me and i agree there is a small bug. My opponent has just resigned and posted a "well done" message. When i viewed the board it appears as if the message was from myself i.e. kingofthe303 says well done.😕