They will run in sync, but independently.

WhatsApp has said that they are working on an update that will allow users to be logged into the app on 4 devices at the same time. This new feature means that users can run WhatsApp on a tablet, smartphone and PC, independently. At the moment, you need to make sure your phone is connected to the internet to use WhatsApp Web. Although this was confirmed on the WhatsApp Beta testing forum, no official release date has been given just yet.