The first thing you have to remember is that your mover did not provide you with "insurance".
Regardless if you purchased additional "valuation" from your mover or elected to just move with the standard 60 cents/lb./article...you do not have "all risk" coverage.
#1. If your claim is for packing damage, it is incumbent on you to keep all packing materials, so that the mover can inspect them. Failure to do so could negate your claim on a technicality. My advice, if you discover packing damage and your packing was done by the mover. Stop unpacking that particular box and alert the mover.
#2. Movers will tell you that you have no coverage for items you packed yourself. NOT TRUE. Do not let a mover tell you otherwise. If necessary the American Moving and Storage Association, AMSA, www.moving.org has more information on this for you to cite with the mover.
#3. Never waive the Bingo Sheet...it is your best way to prove missing items. On a local move, where the is no inventory it may be hard to prove that you are missing something.
#4. The mover is under no obligation to replace a damaged item. The wording in your contract with them is that they will "repair or replace, at their option".
#5. Do not threaten to sue your mover, particularly a large interstate van line. They have in house legal consel and it will do nothing more than cost you money in the long run...even if you are proved right. There is a provision for arbitration...it is much less costly and truly fair, use it.
#6. Use agencies like the BBB to help effect change. I have personally seen movers bend over backwards so their record with the BBB remains good.
There are many other ways to resolve a claim, these tips are just a small sample. Remember that the claims person you are speaking with is a human being and don't take your frustrations with the moving company out on them...they are only doing their job. The old saying goes...you catch more flys with honey than with vinegar!