After snooping around with a flashlight, I think I found out what the culprit is. It's a little foam-looking door that when closed, allows the air flow to the roof vents, and when open, routes the air to the floor. I took pictures, though I dont know if they'll load up.
On the 03 Expedition, it's the big vent looking thing in the third row seating on the passenger side. If you sat in the third row seat, on the right most seat(looking forward), it would probably be by your right knee. If you look in there, you'll see the little square foam looking door.
The problem that I see is that upon opening, or closing , it doesnt stop. When fully open/closed, it continues to try to open/close, as if something isn't telling it stop. So basically the motor is "tripping." Eventually, it stops, especially if the AC is on full blast and the door is open(floor mode.)
Hopefully this helps? Id like to see if this is the problem with your vehicles. My dad wants to "open" it up tomorrow to see what's wrong. If we do( and maybe fix it) Ill post our findings.
BTW, I did this with full cold and full hot.
This is correct. There is another actuator in there that looks almost identical to the blend actuator for hot to cold. It does change vents from top to bottom and it's plastic gears strip. I have this sound also but am to lazy to change the part as it's a pain in the arse. Cheap though.. I believe it's around 57$