Good trip leaders plan for emergencies before the river trip. Part of this planning is identifying a nearby hospital, researching the run in guide books or via the web, bringing a First Aid Kit, etc. When you are preparing for the shuttle discuss what group gear you have like spare paddles, pin kit, location of first aid kits, survival kits, collapsible saw, decent throw ropes. It also helps to frankly discuss paddlers skills. Who has first aid training, SWR training, who is suitable for lead and sweep positions, how many newbie's are on the trip, etc. If the weather is cold, verify everyone has proper cold weather gear and pack some strategic spare layers. All this preplanning pays off handsomely when a real emergency takes place. Knowing where safety and first aid gear are located saves lots of time.
When an incident takes place, someone needs to take the lead as the incident commander. The above planning will have identified this person and all the assets that are available to them. The incident commander ensures safety is in place where appropriate. They also coordinate various rescue tasks so they can be worked in parallel. If outside help is necessary, the incident commander will organize the message and ensure the "runners" can properly deliver the message. This is absolutely essential for getting the right kind of help in a timely manner.
This happens rarely if you did your planning up front. Sometimes, rivers rise to unsafe levels while you are running them. When this happens, it is imperative that you get off the river and evacuate. Climbing out canyons is tiring and no fun but sure beats flush drowning or ending up is strainers. A small portion of the group may need to self-evacuate due to an injury like a dislocated shoulder. This is where knowledge of convenient hiking or logging trails can be very valuable.