We present an analysis of three interactive installations in public spaces, in terms of their support of engagement as an evolving process. In particular, we focus on how engagement unfolds as a dynamic process that may be understood in terms of evolving relations between cultural, physical, content-related, and social elements of interactive environments. These elements are explored through the literature on engagement with interaction design, and it is argued that, although valuable contributions have been made towards understanding engagement with interactive environments, the ways in which engagement unfolds as a dynamic process remains relatively unexplored. We propose that we may understand engagement as a product of the four above-mentioned elements, and in our analysis we provide concrete examples of how engagement plays out in practice by analyzing the emergence, transformation and relations between these elements.