This paper outlines a first attempt to model the special constraints that arise in language processing in conversation, and to explore the implications such functional considerations may have on language typology and language change. In particular, we focus on processing pressures imposed by conversational turn-taking and their consequences for the cultural evolution of the structural properties of language. We present an agent-based model of cultural evolution where agents take turns at talk in conversation. When the start of planning for the next turn is constrained by the position of the verb, the stable distribution of dominant word orders across languages evolves to match the actual distribution reasonably well. We suggest that the interface of cognition and interaction should be a more central part of the story of language evolution.