Ask the person from the moving company giving you the quote or estimate if they will be present during your move. You will have more continuity if the person you explain your move to and get the quote from is present during the move.
You may have up to four contact people for one company; the person who gives quotes, the owner who usually sets up movers for the job, the secretary who answer all your calls and a foreman to oversee the actual move. With so many people involved in your move, your concerns can be lost between the communications of many different people. Ask for one point of contact, so you will not have to catch everyone up to where you are in your move.