https://lwn.net/Articles/548910/
A summary of talks at a recent conference on test automation, with a bunch of links for people who want to follow up and watch videos.
A major theme was WebDriver, which is an API for automating web browsers. Mozilla presented its work on WebDriver support in Gecko and extensions to WebDriver to allow automated testing of FirefoxOS beyond just the Gecko-powered content layer. Google talked about WebDriver support in Chrome/Chromium, including Chrome on Android. Others demonstrated FOSS software that re-purposes the WebDriver API for testing native mobile applications on Android and iOS.
We use Selenium, which makes use of WebDriver. https://www.mediawiki.org/wiki/QA/Roadmap has more. I believe we are not currently using any of the technologies mentioned to specifically perform automated testing on mobile apps or mobile browsers, although I could be wrong.
Chris McMahon was at this conference and may have more specific "we should do foo" recommendations. :)