I just leave. I don’t announce that I’m not taking the next flight. The airlines are really good at giving your seat away if you’re one minute late on a connection so why should I worry about it? The airline will put someone on standby in your seat whether you tell them or not.
I don’t do it often, but sometimes it just makes sense for me. It can go awry, but just be prepared to adapt.
Definitely take your passport. If your final leg is international, you won’t be able to take the first leg without documentation.