Derived from the act of moving the chains in an American football game when a team gets a first down, this expression describes taking a project to the next step, especially one that has lost its momentum for one reason or another. Example: Frustrated with our lack of progress, our boss finally shouted, "Make a decision today about which one to use, and let's move the chains on this."
When a horse runs, its head moves backwards and forwards alternately - in horse racing, if 2 horses cross the line together the one whose head happens to be going forward often wins and is said to win 'on the nod'.
A photo finish is when two contestants (usually in a race) finish at almost exactly the same time, making it difficult to determine the winner. (The saying stems from the practice of taking a photograph when the winners cross the finish line to determine who was ahead at the time.)