All of the documentation is open source in the docs Github repository.
The browser-based DFU does use Web USB for the USB part, but there’s an implementation of the DFU part here. It’s from a public sample implementation, which I modified to work with Particle devices.
The code that gets the system restore zip files and calls the web DFU code is here. That’s not nearly as portable to other applications, but it shows how to call web DFU, so it may be helpful for that.