With the exception of AA which does rebook across tickets, you are SOL if you don't make it to the location at which your next itinerary starts. It is as though you no showed for any other reason. The ticket generally is cancelled and has no value.

Additionally, most carriers impose a single penalty to make a change on one ticket. If you are on multiple tickets and, for example, need to move your trip forward by a day, you will pay a change penalty for each ticket.

Lastly, because you are not misconnecting, most travel insurance does not cover losses on later tickets.