From personal experience, it will only give you the ban message on the device you were originally banned on. Attempting to post from other devices on the same network will only yield captcha error with no respect to OS or browser used. I have attempted to post from 2 linux machines running Firefox, a Windows computer running Chrome, and a mobile device running Firefox and Chrome. Only the original mentions the ban. Clearing cache has no effect across devices.
Furthermore, and I don't know if this is exclusive to Firefox on Linux, it will only tell you "you are banned" in a dialogue box, it does not give a reason why. I only remember seeing the standard imageboard ban message screen on Firefox once, and even then, the reason field was left blank. I attempted to reach out to the email address in the FAQ, and have been given no response after 7 days. It is hard to use the website if I don't understand why I was banned.