I see no other text engine taking over from the browser. So at least for text, the browser is here to stay, even if it's embedded in a control within a native app.
I think you are missing a point I am trying to make. It is not about what language or plug in is used in the browser, but the fact that people are simply not using the browser at all, and using native applications written in Java on Android, XCode on Apple, and WinRT for Windows. Web applications offer a poor usability story on very small screens, why go to http://www.nytimes.com/ on a mobile phone when they have an app that nets you navigate their content far much more easily?
The web is being taken over by proprietary technologies as it is poor on small screens, and the longer HTML5 takes to be finalised, and ecmascript 6 mean that people will move to developing for all three platforms, or the platform their users use the most.