Damage can occur during the moving process by leaking water lines from your washer or refrigerator.  It is important that you take an active role in looking at these at your move out location and also you new residence. 

At your current location when the washer and/or refrigerator are unhooked from the water lines you should inspect these prior to leaving your house.  Take a towel and dry it off and let sit for a while and then check for any drips coming from them.  Some water valves could be old or damaged and it might take time for them to start leaking.  For sure check before leaving your current house.

At your new residence after they have hooked your appliances up check the any leaks every few hours for the first day.  Sometimes leaks don't start immediately but when the pressure builds they could. 

This easy step that can prevent a lot of headaches and thousands of dollars in water damage.  Movers are not plumbers and do this as a convenience for you, and many movers won't even do it at all.