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As I told you in my previous answer, all you need to do is decide whether the player is moving or standing still by comparing the new position with old position, if the player is moving for example, all you have to do is play the walk animation. As you yourself told that, your character is moving, so if you already know that your player is moving, you can also play the movement animation.
We already have a demo, for which I gave you the URL. In our sample, we only exhchange positions and comparing the positions we decide whether the player has moved our not. If he is moving, we simply play the moving animation. No need to send information regarding what animation is being played because it's obvious that if player is moving we have to play moving animation. You can check out the AnimationController.as script attached to the viking prefab in our sample.
