There is a new and quite interesting paper on this topic, by Kyle Mangum and Patrick Coate:
This paper offers an explanation for declining internal migration in the United States motivated by a new empirical fact: the mobility decline is driven by locations with typically high rates of population turnover. These “fast” locations were the Sunbelt centers of population growth during the twentieth century. The paper presents evidence that as spatial population growth converged, residents of fast locations were subject to rising levels of preference for home. Using a novel measure of home attachment, the paper develops and estimates a structural model of migration that distinguishes moving frictions from home utility. Simulations quantify the role of multiple explanations of the mobility decline. Rising home attachment accounts for nearly half of the decline, roughly as large as the effect of an aging population, and is consistent with the spatial pattern. The implication is recent declining migration is a long run result of population shifts of the twentieth century.
For the pointer I thank the excellent Tyler Ransom.