Overbooked flights happen all the time on all three of the legacy carriers. Most of the time these issues resolve themselves naturally and passengers are never the wiser. Some folks will cancel their ticket, no-show, or switch to a different flight. Sadly, others will misconnect. In all of these cases, nothing needs to be done as the problem will solve itself.
Sometimes, however, the oversold situation persists and there are literally more passengers in the gate area than the flight can accommodate. When this happens, the agent solicits volunteers willing to take a later flight in exchange for compensation. In rare cases where they get no takers, they may have to involuntarily deny boarding (IDB) to someone.
Once in a blue moon, all of this will break down and they’ll try to deny someone boarding who has already boarded. As ridiculous as that sounds, it’s exactly what happened to Dr. Dao on his United flight from Chicago to Louisville. We all know how how that ended.