Abstract: In biology, natural selection is the main explanation of adaptations and it is an attractive idea to think that an analogous force could have the same role in cultural evolution. In support of this idea, all the main ingredients for natural selection have been documented in the cultural domain. However, the changes that occur during cultural transmission typically result in convergent transformation, non-random cultural modifications, casting some doubts on the importance of natural selection in the cultural domain. To progress on this issue more empirical research is needed. Here, using nearly half a million experimental trials performed by a group of baboons (Papio papio), we simulate cultural evolution under various conditions of natural selection and do an additional experiment to tease apart the role of convergent transformation and selection. Our results confirm that transformation strongly constrain the variation available to selection and therefore strongly limit its impact on cultural evolution. Surprisingly, in our study, transformation also enhances the effect of selection by stabilising cultural variation. We conclude that, in culture, selection can change the evolutionary trajectory substantially in some cases, but can only act on the variation provided by (typically biased) transformation.