User testing is the process of evaluating a web application by testing different parts of that site to assess its viability on the web. Different tests can be run to check a website's functionality, accessibility, and browser compatibility, just to name a few (QATestLab.com).
Before any cross-browser testing takes place, you can establish a baseline by testing your site on its primary browser, which is used as a point of comparison versus other browsers. Next, you can perform a site traffic analysis to determine which browsers are the most important to test.
The tests themselves can be done with emulators or simulators, however those are less reliable for mobile devices, and not as easily scalable. Tests can also be done by cloud-based platforms or by companies with their own device labs (BrowserStack.com).