There are two ways to miss a connecting flight. Obviously they lead to two different outcomes.
The first way is yelling at the gate personnel who is showing you the plane already rolling on the tarmac. Questioning their professional integrity and promising never to fly again on the same airline. Keeping on yelling for 10 minutes and then, when cooled off, ask for further instructions. At that point walking 60 gates back to the service center, and getting rebooked, eventually.
The other way is realising that the plane is gone already, smile at the personnel, and kindly ask for an alternative option. Apologizing for the jerk yelling at the other desk, and getting rerouted immediately on the next flight. Also getting a nice compensation to spend on food and drinks while waiting for your next flight.
Being nice goes a long way.