How it works

WHAT YOU SHOULD KNOW TO UNDERSTAND SPECIFIC CONTENT FOR MOBILE.

DEFINITIONS.
Browser. The program you use to visit a web page (e.g. Google Chrome, Firefox, Safari…).
Server. Hardware + software that provides functionality for other programs or devices, called clients (browsers and client devices). The files of your website are stored on the server. On the same server runs the software to build the pages of your website.
PHP. The programming language that runs on the server.
HTML.Markup language interpreted by the browser.
CSS. The language used to style the HTML document and define how the HTML should be displayed in the browser.
Responsive design. The approach to web design that makes web pages render well on a variety of devices and window or screen sizes from minimum to maximum display size. The responsive design is made with CSS.

HOW IT WORKS (simplified, not exhaustive, and with some imprecisions to be clearer).
When you visit a web page of your website, the server runs the so-called PHP to build that page. Before the webpage is ready, WordPress calls all the files of the core, plugins, and theme and gets all the information from the database.
The content of the pages is stored in the database.
When the page is ready the server sends it to the browser and you can see it.
What you see on your browser depends on how the page is in terms of structure given by the HTML and the style given by the CSS.
The CSS is interpreted by the browser.

Specific Content For Mobile does nothing else than replace the content of a web page with the content of the mobile version. It works only on the server. After the page is ready, and the server sends it to the browser, Specific Content For Mobile has no control of what happens on the browser.
The code of Specific Content For Mobile runs only on the server. It detects if the device is mobile or desktop and if you have defined a mobile version, it replaces the content, nothing else.

How the page will appear on the device will depend on the CSS and responsive design.

Reading this post it should be clear why it’s so important that the caching system distinguishes between mobile and desktop. Read this thread for more details: https://www.ads-software.com/support/topic/the-caching-plugin-if-any-must-distinguish-between-mobile-and-desktop/.