The migration of people between different cultures has affected cultural change throughout history. To understand this process, cross-cultural psychologists have used the 'acculturation' framework, classifying 'acculturation orientations' along two dimensions: the willingness to interact with culturally different individuals, and the inclination to retain the own cultural identity ('cultural conservatism'). Here, using a cultural evolution approach, we construct a dynamically explicit model of acculturation. We show that the evolution of a multicultural society, where immigrant and resident culture stably coexist, is more likely if individuals readily engage in cross-cultural interactions, and if resident individuals are more culturally conservative than immigrants. This result holds if some cultural traits pay off better than others, and individuals use social learning to adopt more advantageous cultural traits. Our study demonstrates that formal dynamic models can help us understand how individual orientations towards immigration eventually determine the population-level distribution of cultural traits.