The driver is the main guy in charge. Everyone works for him and he's responsible for any claims for damage. If stuff happens you can work with him and he will potentially cover stuff on his own to prevent a claim, assuming it's not a huge claim. He may very well be the only guy on the packing and moving crew that is there for the delivery.
.
During one of our moves that was several states away the guys damaged a wall with a couch after t told them it wouldn't go thru that way. I was coming back to the house for a while so I told the driver I would deal with the damage.
When we got to our new house the downstairs was either tile or glue down hardwood that was in new condition. I stressed that NO Boxes were to be placed on the hardwood. My wife was directing things inside and I was dealing with the checkout list and directing outside. Apparently the guys could not figure out how to assemble a bed upstairs and asked my wife for help. When she returned the other guys had placed about 8 stacks of boxes right on the hardwood. Our previous driveway was gravel and the loaders staged boxes around the truck and rocks got embedded in the cardboard and they used the kick and twist method of pulling the hand truck from under the boxes on our hardwood floors.
Several days later when my wife was unpacking those boxes we found the floor all scratched up and submitted a claim. The driver was quick to deny any damage saying there was no way it happened. Adjuster first told us we were basically screwed because the drivers word was better that ours. After the adjuster dug deeper into the paperwork he found the driver did not have us sign the release for damages on the delivery side so they had to side with us. We received a check for $10,000 because they had no way of matching the flooring.
If the driver and the company had not been so quick to try and deny our claim we probably would of been happy with $500.