While existing theories of political polarization tend to suggest that the opinions of liberals and conservatives move in opposite directions, available data indicate that opinions on a wide range of moral issues move in the liberal direction among both liberals and conservatives. Moreover, some political scientists have hypothesized that this movement follows an S-shaped curve among liberals and a similar, but later, S-shaped curve among conservatives, so that polarization on a given issue first increases (as opinions at an initial stage move faster among liberals) and then decreases (as opinions at a later stage move faster among conservatives). Here we show that these dynamics are explained by Moral Argument Theory, a cultural evolution theory positing that opinion shifts on moral issues arise from a certain content bias in social opinion transmission. This theory also yields several other specific predictions about trends and polarization in moral opinions, which we test against longitudinal data on 55 moral issues from the General Social Survey (sample sizes between 1798 and 57,809 per issue). The predictions are generally confirmed. We conclude that a cultural evolution perspective can provide valuable insights for social science in understanding contemporary societal changes.