My measure of mobility is the correlation between these perceived ranks. A lower correlation means a weaker connection between position growing up and position today: that is, more mobility. I'll say "high mobility" rather than "low correlation" because it seems easier to keep things straight that way.
The United States ranks tenth in perceived mobility, with a correlation of .431. Most of the countries that have the highest perceived rates have made a transition from Communism: the top three are Latvia, Croatia, and Ukraine. New Zealand is just ahead of the United States, and Austria is just behind.
The lowest two are South Africa and Israel. Portugal, Italy, Finland, and Great Britain also rank low.
Note that the correlation just doesn't involve changes in the averages. For example, Ukraine and South Korea have similar correlations (.347 and .360). But in Ukraine, the average current position is 3.54 and the average position when growing up is 4.59; in South Korea, the averages are 4.51 and 4.27. That is, most Ukrainians think they have moved down, while most South Koreans think they have moved up. The United States is is the middle here: the average for today is a little higher than the average when growing up.