How and why are moral norms for various issues changing across the globe? Moral argument theory uniquely addresses these questions by positing that moral norm change is driven by 'individualizing' arguments: concerns about harm, fairness and liberty. We test this theory in a preregistered study of the arguments for both sides of 33 moral issues for which the global directions and rates of norm change were estimated in available longitudinal survey data from 94 societies. We also use available cross-national data to estimate the extent to which each society relies more on individualizing arguments than other kinds of arguments. In support of the theory, norms' justifiability by individualizing arguments was found to predict their global change, and the effect of individualizing arguments on norm change is stronger in societies that rely more strongly on such arguments. These findings demonstrate a fundamental pattern in the contemporary cultural evolution of morality and highlights the key role played by individualizing arguments.