Put a bookmark in your bookmark toolbar with the address below (copy and paste it). Then if you are viewing a game that has a URL ending in &nmove=..., clicking the bookmark will advance to the next move.
javascript: (function(){var s = window.location + ""; var i = s.search("&nmove=") + 7; var mn = parseInt(s.substring(i)) + 1; window.location = s.substring(0, i) + mn;})()