The assumed architecture model is problematic. You shouldn't have an "Aegir hub" server separately from Dev, Staging and Prod servers; you should simply have Aegir Dev, Aegir Staging and Aegir Prod servers.
To move sites from one Aegir environment to another, you would remotely import them.
For details on this topic, see Aegir DevOps: Deployment Workflows for Drupal Sites.
For updating code on whichever server, you would navigate to Platforms » Add Platform. From there, you would choose a deployment strategy involving Git. See the Deployment Strategies documentation page for more information.