Drones pick up their cargo at a supply depot, but then they drive against that depot for sometimes more than 30 seconds before they leave again to deliver their cargo.

Can you provide a save game?

general hint on connecting roads to depots:
make the road at least two tiles wide, and have a larger area of road tiles around the "caution" tile. drones sometimes have difficulties to quickly pathfind their way before "docking" or after "undocking" when the depot is only barely connected with a 1 tile wide road. best to have 2 or all 3 sides of the caution tile covered with roads.