We develop a minimalist compartmental model to study the impact of mobility restrictions in Italy during the Covid-19 outbreak. We show that, while an early lockdown shifts the contagion in time, beyond a critical value of lockdown strength the epidemic tends to restart after lifting the restrictions. We characterize the relative importance of different lockdown lifting schemes by accounting for two fundamental sources of heterogeneity, i.e. geography and demography. First, we consider Italian Regions as separate administrative entities, in which social interactions between age classes occur. We show that, due to the sparsity of the inter-Regional mobility matrix, once started, the epidemic spreading tends to develop independently across areas, justifying the adoption of mobility restrictions targeted to individual Regions or clusters of Regions. Second, we show that social contacts between members of different age classes play a fundamental role and that interventions which target local behaviours and take into account the age structure of the population can provide a significant contribution to mitigate the epidemic spreading. Our model aims to provide a general framework, and it highlights the relevance of some key parameters on non-pharmaceutical interventions to contain the contagion. © 2020, The Author(s).