Advanced Version Control Architecture - How to maintain multiple installs of a website/application for clients

I have a question that combines multiple concepts.

I do a lot of work with real estate agents and I am working on building a complete solution using Laravel where we build one awesome framework for a website, but allow clients to add style touches such as color scheme, logos, etc.

My question is how do you use a version control system such as Git so that when you push files to your master branch for the core files, all of the production websites you have deployed are updated.

We may have 20 clients running the application on different servers. How can you set up some sort of "call back" that checks for updates to the version of the core files and merges those with the production applications of those various clients. I am thinking about something like Wordpress where your core files are updated automatically from the main repo.

I imagine there is some sort of listener class that looks back to the main repository on a timed interval to check for a different version, but, this is just a guess.

I am open to any ideas, concepts, code examples, resources, or anything that could point me in the right direction on how to accomplish this.