Before filiing a pull-request you should check that the changes introduced by you still make the site build. Therefor, you should always build and test locally before you ask for a pull request.
How to automatically update the documentation locally during development:
$[~/PROJECT_ROOT] sphinx-autobuild content/ _build
The static webpage is hosted on Github and automatically build be pushing changes to the main branch. The build web-page is displayed as the stuff build on the gh-pages branch. This process is automatic. A guide to the principles can be found here.