What happens at connections when flying standby from the US?

Each airline has its own processes for standby non-rev passengers. Ask in advance what those are. Usually you'll need to put yourself on a list for each flight of your trip. The list of all non-rev passengers will be stack ranked using a number of factors like seniority and time you signed up. Then you have to show up to the gate, wait and see if there's enough free space to make it down the list to your name shortly before departure.

Normally, the gate agent is boss of this process; when in doubt ask them, but remember they have paying customers, and an expensive crew and plane to send off on time, to prioritize over you.

Some airlines will send your boarding pass directly to your phone and show your name as cleared on a screen in the boarding area once it's been determined that there is enough space. When you don't see those, normally the gate agent will just call your name as soon as they are confident that there is a seat available. If the flight obviously has room, the gate agent will likely call you before the boarding process. If the flight is closer to full, they will wait until most of the way through the boarding process to see how many passengers actually show up. If they don't have space for you they'll ask you to sit and wait while the complete the paperwork and get the flight out, after that they'll help you figure out what your next option is.