As long as all your flights are on the same ticket, you will be protected in case of delays.
The rebooking process is frequently not automatic, however. As soon as you arrive off a flight and can't make the next one, you need to approach the customer service of the carrier who just took you to your current location. They will be responsible for reorganising the rest of your journey. The rebooking should be completed free of charge; if you are inconvenienced to a greater degree (e.g. last flight of the day, very long layover, etc.), make sure you check or think about negotiating additional rights for your inconvenience, within reason.
If you arrive at your location but your ongoing flight hasn't actually departed yet, try to get in touch with the onward carrier; they may be able to expedite your passage, hold the flight (unlikely), or otherwise help you (even if this means you need to go back to the arriving carrier). If you have truly misconnected (i.e. the onward flight has already left), then go to the carrier that just got you into the port.
Happy to have others weigh in on this, especially the part about which carrier has the responsibility at the location of the misconnect.
