I also manually export my Mirth configurations and manually commit them to a Git repo. This works for me because I'm part of a very small team that works on a single Mirth instance. My sense is that this is the common practice in the Mirth community. That said, there are a few other options available.
There's a Channel History extension for Mirth Connect that allows you to compare different revisions of your Mirth Channel configuration. It's one of the standard extensions that NextGen Healthcare advertises for Mirth. You can find out more about it here: https://www.nextgen.com/products-and-services/integration-engine?extension=none
There's also a 3rd-party command-line interface tool from SagaHealthcare that is supposed to allow automagic integration with Git and SVN repos. According to the documentation, it integrates version control of code templates, channels, configuration maps, and global scripts. You can read more about this one here: https://github.com/SagaHealthcareIT/mirthsync