Custom theme – versions and update

For several years I do and maintain the website of an NGO where I volunteer. Due to the needs of the organization have many subdomain. But when I need to update the code of these domains should be me and go through a lot of files, it takes valuable time.

Now I read about network of my sites (Multiple domains with multisite), but I still need to know:

How I can make a changes of this theme in one place and in all other sites to show “There is a new version of XXX available. View version 1.X details or update now.” ?

Please help.
Thanks