What happens when you miss a connecting flight because of the airline?

You would have to go to the gate agent in the airport and he or she will tell you what they can do. They good airlines will give you a hotel and maybe even a voucher for a free flight. Unless they can put you on another flight within a reasonable time, they will most likely put you in a hotel. I'm uming you are flying United, since you have a connection at Dulles. Last time I flew on united, I had no problem, but they seem like a very good airline. Just go to the gate agent to find out what you can do.