We develop a multiple-events model and exploit within and between country variation in the timing, type and level of intensity of various non-pharmaceutical interventions to study their dynamic effects on the daily incidence of COVID-19 and on population mobility patterns across 135 countries. Taking into account the contemporaneous presence of multiple interventions, we remove concurrent policy bias from the effect of each policy of interest, and we establish that policies curb the epidemic by changing population mobility patterns in a manner consistent with time-use and epidemiologically relevant considerations. We are thus able to shed light on the mechanisms through which confinement measures contribute to "flattening the curve".