Roy -- Oh, Chrome only shows it on the network tab, not the console. Firefox shows it in both locations.
Roy -- If you open up the console in your web browser's developer tools, you'll see all ajax requests made as you interact with the page. Just make those requests manually with PHP.
Devourer -- I can use php's file_get_contents(URL) to retrieve the html document given by that server. Some servers nowadays just load very few at the beginning and load stuff via AJAX functions. How would I try to get these "later datasets" via PHP?