The social and cognitive mechanisms of cultural evolution have been studied in detail for different domains: language, technology, the economy, art, etc. However, a model that incorporates the function of a cultural tradition and that is able to compare evolutionary dynamics across cultural domains has not been formulated. By exploring the dynamics of comparable linguistic, technological, and artistic experimental tasks, we test the effect of domain-specific function on evolutionary mechanisms such as inheritance, innovation, and selection. We find evidence that cultural domain shapes both the structure of the traditions and the way the cultural-evolutionary mechanisms operate. The simplifying effects of cultural transmission are noticeable in language and technology, but not in art; innovation is highest in art and lowest in language; and functional pressures lead to different morphological adaptations across domains. This speaks of a crucial role of function and domain in the evolution of culture.