What happens if you book tickets through an airline or travel agent, but on the travel date miss a connecting flight (either because the first flight was delayed, or because immigration/security took too long at the connecting airport)? Will you be put on the next flight for free? Are you expected to pay for a new ticket or a change fee? I'm specifically wondering about Continental/United, but also curious about airlines in general.
Generally, if you have a through booking, then if you miss the flight because of the airline's fault then you become their problem.
This doesn't mean you'll automatically get put on the next flight. You'll need to talk to the airline and ask for help, and ask for re-booking. If the next flight has spare seats, then all should be well. Most likely though, you'll get put on standby for the next flight, and hopefully will make the next one or the one after. As it's the airline's fault, you should get meal vouchers, help with a hotel if needed etc. (Note - the exact level of help you get depends on the jurisdiction you're in, and the home jurisdiction of the airline)
However, if you miss the connecting flight because of your own fault (eg you fall asleep while waiting near the gate), then it's probably no different to if you have two independent bookings....