Drupal login using other subdomain user account with certain role
Are you using different databases for all sites? or the Domain Access module? https://www.drupal.org/project/domain

If you are using different databases for each site and if they are on the same server I would consider to let the sites share the users + users_roles table. 
In that way all users exist on all of the sites with the same password and have the same role everywhere. You can then assign different permissions to different roles per site.