After more than 1 year into the COVID-19 pandemic, governments worldwide still face the challenge of adopting non-pharmaceutical interventions to mitigate the risks posed by the emergence of new SARS-CoV-2 variants and the lack of a worldwide equitable vaccine allocation. Thus, it becomes crucial to identify the drivers of mobility responses to mitigation efforts during different restriction regimes, for planning interventions that are both economically and socially sustainable while effective in controlling an outbreak. Here, using anonymous and privacy-enhanced cell phone data from Italy, we investigate the determinants of spatial variations of reductions in mobility and co-location in response to the adoption and the lift of restrictions, considering both provinces and city neighbourhoods. In large urban areas, our analysis uncovers the desertification of historic city centres, which persisted after the end of the lockdown. Such centre-periphery gradient was mainly associated with differences in educational attainment. At the province level, the local structure of the labour market mainly explained the variations in mobility responses, together with other demographic factors, such as the population’s age and sex composition. In the future, targeted interventions should take into account how the ability to comply with restrictions varies across geographical areas and socio-demographic groups.