Terri A. Sexton (UC-Davis & California State University, Sacramento) has published Proposition 13 and Residential Mobility, 50 State Tax Notes 29 (Oct. 6, 2008).  Here is the Conclusion:

Empirical studies confirm that households respond to increased transaction costs by not adjusting their housing consumption as often as they would in the absence of those costs, thereby reducing their mobility and introducing inefficiencies in housing markets. In the case of acquisition value taxes, however, the empirical evidence is mixed. In California the consensus has been that Proposition 13 has had a significant negative effect on mobility. However, the estimates of its magnitude vary from study to study and are shown to vary over time and geographically across the state. In the two other states that have been studied, Florida and Georgia, the empirical evidence suggests that the lock-in effect, if it exists, is not significant.

The efficiency consequences of reduced household mobility include the excess burden on households from suboptimal housing consumption, inefficient labor market outcomes, longer commutes with associated environmental and congestion costs, a reduction in the supply of smaller homes for young and old home buyers, and reduced incentives for households to vote with their feet, thereby impeding the efficient provision of local public goods.

Theory suggests that an acquisition value property tax imposes a moving penalty similar to other costs associated with selling one home and moving to another, such as higher mortgage interest costs, capital gains taxes, and real estate transfer taxes. However, more research is needed to ascertain the magnitude of the moving penalty and its associated efficiency costs.