How to implement automatic updating of running projects?

Question

Online shopping

There are several online stores that are created on a certain engine assembly with modules and other goodies.
From time to time, there is some kind of bug or functionality improvement. If the issue with the preparation of the site is solved simply. We use a repository in GIT and all new sites are deployed from an updated distribution.
What about existing sites. I would not want to change this manually in every site.
It also adds to the complexity that stores have their own "files" because of the designs, so I want each "type" of the site to "pick up" their own files from the server.
What are the suggestions?

Answer 1

If all sites have the same "own files" - git submodules. If the difference is large, then separate them into branches, where everyone inherits from the master with common sources, but each has some peculiarity. Make general changes in the master and push to branches, either manually or automatically via git hooks, for example.

Answer 2

Initially, make the right architecture. It is
necessary to allocate a common part, as large as possible - for all sites. And they would update it at the same time (or in general it would be a single backend for everyone).
And unique pieces for each site.
And knowing exactly where the border between them is, it would be easy to update at least every hour.
Git doesn't solve any of this. Git is just a basic tool with which you can track changes.