I'm using an external battery on a shoulder rig with an EX1R. If there's also a SONY BP battery in the camera's battery slot, anyone know which will drain first, the internal or external source?

External. When there is power connected to the DC IN, it will drain that over the BP-U battery. Do be warned that there is no low voltage switchover when you do this so you must manually yank the DC IN to switch to the BP-U battery.