There used to be a "park" feature on hard drives (years and years ago) which you had to use in order to move the machine from one side of the desk to another. But, thankfully, those days are over.
Computer components will sometimes fail simply by being powered off and back on. If for some reason the power in the new location is "dirty" or in some other way inconsistent that could also be a cause of failure. Low voltage or high voltage conditions do play havoc on electronics.
Also, hard drives do get old and need replacing every so often. My rule of thumb is to keep a drive no longer than 3 years. After that power cycles have a habit of killing them. Sure, the drive manufacturer might claim a 5 year or 10 year MTBF; but those numbers usually involve extremely few power cycles. Ever noticed how server drives always have higher MTBF numbers than desktop? It's not because of quality differences.
Motors fail, magnetics fail, etc. I'm not defending the handling by the moving company (obviously I wasn't there so I have no idea); however, I do just want to point out that there are other factors.
This is why server parts, especially mechanical ones, have a higher incidence of failure in the first month than any other time. Typically, a server is put together and power cycled many times during that first month due to software installations and other set up tasks. Once in it's "home" most servers are simply not shut down very often. Rebooted? Yes, but not completely powered off and later turned back on.
This means the mechanical parts rarely have to go through the most stressful part of their life which is shut down, park heads, and start up.