We analyze mobility changes following the implementation of containment measures aimed at mitigating the spread of COVID-19 in Bogot\'a, Colombia. We characterize the mobility network before and during the pandemic and analyze its evolution and changes between January and July 2020. We then link the observed mobility changes to socioeconomic conditions, estimating a gravity model to assess the effect of socioeconomic conditions on mobility flows. We observe an overall reduction in mobility trends, but the overall connectivity between different areas of the city remains after the lockdown, reflecting the mobility network's resilience. We find that the responses to lockdown policies depend on socioeconomic conditions. Before the pandemic, the population with better socioeconomic conditions shows higher mobility flows. Since the lockdown, mobility presents a general decrease, but the population with worse socioeconomic conditions shows lower decreases in mobility flows. We conclude deriving policy implications.