So I did a basic analysis using the General Social Survey, which asked people how much schooling they had, and also how much their father had. I made a tables of father's education by own education for six groups: born in 1909 or before, born 1910-19, 1920-29, 1930-54, 1955-69, and 1970-90. Both were classified into seven categories: 6th grade or less, 7th or 8th grade, some high school, high school graduate, some college, college graduate, and graduate education. I restricted this to white men because it seems pretty likely that the trends will be different for women and non-white men, and to people age 25 or over because many of those below that age are still in school.
I fit a model that described the changes with a single number representing the difference from the last cohort. The number represents the association between father's and son's education, so high numbers mean less mobility. Here are the results as a figure:
The conventional wisdom seems to be right: the association is substantially higher in the last generation (people born 1970-90), although there was some upward trend before then. The scale of the vertical axis isn't meaningful, but in a rough way you could say the association is about twice as strong in the youngest cohort as in the oldest.
*I mean those in relative terms--how much education a person has relative to what other people in their generation has. In an absolute sense, upward movement has been much more common, because average levels of education rose from generation to generation until the 1970s.