What are my rights if I miss a flight connection due to the first leg being late. Tickets are on the same airline booked at the same time?

What are my rights if I miss a flight connection due to the first leg being late. Tickets are on the same airline booked at the same time?

Unless the delay has been caused by mechanical reasons (pretty rare) you really don't have any rights other than you get the next available seat on the next flight out. You would still be responsible for food and any extra hotel or travel costs (unless it's mechanical). Most delay's are caused by a combination of weather and air traffic control. Weather is a big one since it could be weather in the city the plane is coming from, your city or your connecting city. Regardless there's certainly no monetary incentives for even delayed mechanical flights other than food vouchers, and a free hotel if needed.

The answer depends on which legislation is of application to your flights.

Namely air passengers are better protected in the European Union than in the USA in terms of compensation, re-routing, and assistance from the airline.

Compare:
Europe:
(see FAQ: in your case, you are entitled to choose between different options).

USA:
(basically, you have to negociate with the airline)…

This can be rather limited depending on the choices you made when booking. If you went against a recommended connection time, you find it hard to defend if the incoming flight was delayed unexpectedly but by less time than would affect the standard recommended connection time.

I.e.
Flight 1 arrives 11:00am from a scheduled time of 10:30, and flight 2 departs 11:30am. That would be too tight connection, even on time flight1 it would be tight if terminal changes, etc. Were required.

If flight 1 arrived 11:00am from scheduled 10:45am and flight 2 departs 12:30, you could be vindicated for missing it with just 15 minutes lose.

Personally, you would need to speak nicely to the desk staff, to get agreement on what you could be transferred to depending on the booking terms and conditions.

The carrier is responsible for getting you to your destination.

If the airline is at fault for you missing the flight, they have to book you on to the next available flight.

They will get you on the next available flight to your destination; available is the key word there. If it's a busy route, you may find that you have to wait for an open seat. They are not obligated to put you on another airline, although some will work with a partner/alliance airline to get you there faster if possible.

Depends on what country you are in.
Basically the Airline will get you to destination.
They will try first their flights and may send you in some other direction to get you to final destination.
They might offer you a meal voucher for a short wait and might offer you a hotel room if it is a long wait. YOU DO NOT get the luxury suite with unlimited room service.
YOU DO NOT get magic next flight out. You get next available seat.
YOU DO NOT get a FREE unlimited Hotel stay or unlimited grazing in the food court.

As every other passenger on the flight most likely also needs some change in their plans have a little patience with the Agents rearranging your and hundreds of others next few hours.

You might some points added to your program if you have one, they may give you some sort of voucher/credit to use on a future date. If you are patient and nice to the person dealing with the situation they tend to do a little more to assist you.
They know you are upset with the change in plans. They are not the pilot they are not the weather gods. They do not have some magic plane to come assist. They are trying to help you and all the others alongside you.
It happens you deal with it at the time. YOU HAVE NOT been left stranded at the side of the runway.
Keep your cool and you will be helped being nice to the Agent assisting can get you a little extra.
It does happen a large storm disrupts hundreds of flights tens of thousands of passengers will have the same issues. It takes time to rearrange new flights (seats) for all.
The delay to you may be an hour or two or days depending on situation.

Suggest you take some time and actually read that fine print you agreed to when you purchased your ticket.
Every Airline does not have the same compensation. Every Airline does not have the same ability to access other flights.

European passengers have a much better compensation package by law than North American travellers.

IF you booked your itinerary of flights as a package the Airline staff at the change place already know you will be late and have started working on some alternate arrangement before you even arrive. All of the other passengers may have missed connections too.Agents will greet the plane and direct you to the alternate flight arranged for you or to some place at that Airport they have set up to deal with the situation. EVERY AIRPORT does not have exactly the same counter arrangements, help desks, information counters.

The Airline check in agents are not the weather GODS, they are not plane mechanics they are not hotel booking clerks, they are not credit card refund officials. They are not flight dispatchers they are not your sandwich maker or coffee barista.
Keep your cool let them do their thing to assist you, They have no desire to have you wandering around the Airport any longer than they have to. They also like to go home.

Generally they'll put you on the next available flight with seats. Beyond that, none.
Keep in mind that some destinations may only have 1 flight a day so missing that flight could be a long layover.
If you're polite and cordial, they may offer a discounted hotel but for missed connections, they're not required to.