I spent 20 years in the military and moved every three years. I moved all over the west and to Florida. The movers moved my guns everytime, or at most of my guns. I always kept three or four out for use along the way hunting with my Dad, Brothers, or friends. If the movers take them make sure that every one is on the inventory by make, model, caliber, and serial number. Never let the driver say, I'll take them, but we don't want them on the inventory. (that's a sure sign he is going to steal them) If they are not on the inventory you have no proof that he ever had them.
My wife is a claims examiner for the federal government, her job is to reimburse families for damaged or stolen items by movers. Guns and Prints are the main items that get stolen. The drivers tell the customers that if the item goes on the inventory it tells a theif where and what the item is for them to steal. Well it sounds good to the customer, but like I said if it's not on the inventory it never got shipped.
What I would personally do. Forget the movers, lock the guns in gun cases, or put them in boxes like they came in and tape them shut. (You can sometimes find them at gun shops, or at U-haul). Then pack them under the bottom of all your stuff you are packing in the car or truck. Take your time, don't do anything to get stopped, and just be your normal self. Even if you get stopped there should be no trouble unless drugs are found in the car. Then the whole world will fall apart.
I just finished a stint on the Grand Jury, a young fellow got stopped, his girl friend had a small amount of grass in her purse. While I don't agree with smoking pot, it's not a big deal here in Alaska. The problem came when the Troopers found his pistol and rifle in the vehicle. They were in the trunk in a locked case, but that did not matter. The mear fact that guns were present with the grass was enough to get him locked up, and turn it from a misdmeaner to a felony. He will loose the guns, his car, and possiably spend time in jail.
