The U.S. is a big place and locations with crazy housing costs are not the norm. I would think that most people moving for a job would already have a job available. I suppose there are plenty of cases where people might move without a job offer in advance because anything would be an improvement.
Maybe today people don't move as often simply because EVERYTHING is getting more expensive while the jobs people could land are staying the same or getting worse. Moving is quite expensive, especially when you don't have a job and especially when you don't have a job lined up at the place you are moving to. Gas is expensive for moving long distances. Getting into a house or an apartment is expensive. How many jobless people have the savings to withstand all this?
It would also be interesting to see more statistics. For example, where were people moving to and from when the rates were the highest? What industries were supporting the places these people moved to? Maybe the decline of jobs that could take just about anyone, in large numbers and paid well (manufacturing) contributed to the decline in mobility. There really isn't anything which could take the place of engineering.
Though people may be less mobile today, the options for that mobility will always be great. The U.S. is a huge nation with a lot of diversity. You can't really compare mobility in a nation like the U.S. with 50 states to a small country like that of Denmark or Sweden.