For local automated testing of mobile browsers on both iOS and Android you probably want to have a look at https://appium.io/, which uses the Webdriver protocol.

If you want to run the same tests against both mobile and desktop browsers, I recommend https://webdriver.io/, which uses Appium for mobile testing.

If you're interested in a Docker setup as a boilerplate to test against all major browsers (both mobile and desktop) with best practices for e.g. test video recordings, you might also wanna take a look at this repo of mine: https://github.com/blueimp/wdio