Is it just me or "Live Demo" actually doesn't send any AJAX requests?
I've used pjax with Laravel before, and it's trivial. Kinda does the same thing.
There is a Laracasts video explaining it here: https://laracasts.com/lessons/faster-page-loads-with-pjax
Demo page here: https://pjax.herokuapp.com/
you can click on the code tabs and see what's going on, and it should be easy enough to map to whatever backend technology you are using