On a 97 Firebird what exactly happens mechanically or electrically to shut off the motors in the headlights when they go to the down position?

With noise yours is making the plastic gears are worn out. Check your earlier question for a link to how to replace and repair the gear with metal gears

There is a module in the circuit that runs the motor up to 6 seconds. When it senses a high current draw (the motor cannot move the door any further) it shuts the current. It's a terrible system... as you probably know. You can replace the motors, or change the gears to metal, but there's nothing you can do about the module. http://transamcountry.com/community/index.php?topic=39600.0;wap2