In the last decades, the notion that cities are in a state of equilibrium with a centralised organisation has given place to the viewpoint of cities in disequilibrium and organised from bottom to up. In this perspective, cities are evolving systems that exhibit emergent phenomena built from local decisions. While urban evolution promotes the emergence of positive social phenomena such as the formation of innovation hubs and the increase in cultural diversity, it also yields negative phenomena such as increases in criminal activity. Yet, we are still far from understanding the driving mechanisms of these phenomena. In particular, approaches to analyse urban phenomena are limited in scope by neglecting both temporal non-stationarity and spatial heterogeneity. In the case of criminal activity, we know for more than one century that crime peaks during specific times of the year, but the literature still fails to characterise the mobility of crime. Here we develop an approach to describe the spatial, temporal, and periodic variations in urban quantities. With crime data from 12 cities, we characterise how the periodicity of crime varies spatially across the city over time. We confirm one-year criminal cycles and show that this periodicity occurs unevenly across the city. These `waves of crime' keep travelling across the city: while cities have a stable number of regions with a circannual period, the regions exhibit non-stationary series. Our findings support the concept of cities in a constant change, influencing urban phenomena---in agreement with the notion of cities not in equilibrium.