There are almost no company long distance movers anymore. Movers are small businesses that sign on with a major carrier and receive their work from them. So the "Bekins" mover you use will probably not be the "Bekins" mover a customer from another state uses.
Saying that, I used a company recommended to me by Villages Relocation. They recommended Stevens Worldwide Van Lines, which was represented by Cousins USA Moving & Storage out of Ft. Lauderdale, Fl.
I scheduled a couple more estimates with competitive movers in addition to Stevens just to make sure their quote was in line and then went with them and was the recipient of a flawless move. The Company was quick to thank me for my business and to ask me to let The Villages Relocation people know how I rated their service. I would definitely recommend this company, if they serve your departure point.
http://www.cousinsusa.com/