Actually, just after posting the above, I had the privilege of watching the operation actually happen here at LAX T4. The air bridge started moving towards the plane as the pilot was turning off the engines. The air bridge operator (or gate agent) opened the door 36 seconds after engine shutdown.

I do note that they only attach at the bottom of the air bridge - they don't bother folding the top bit out to create a cover, so maybe weather-dependent (although it was -8C in Boston yesterday morning and I remember the exposure to the freezing cold, so perhaps they only do ths in heavy rain).