How do I treat injuries in a caravan?

You can make temporary beds and treat the wounds like you normally would if you were at the colony with a bed (aka having another person treat the wounds when they are in a medical bed).

(around 9:00 we can see this in the Alpha 16 preview)

Two very accessible, but not immediately obvious ways:

• Under Architect > Furniture, place a Sleeping Spot
• If you added Bedrolls to your caravan, grab one and place it on the ground

Since a bed is required in order to treat injuries, either option will allow another colonist to tend to injuries.