We present some empirical evidence on the dynamics of price instabilities in financial markets and propose a new Hawkes modelling approach. Specifically, analysing the recent high frequency dynamics of a set of US stocks, we find that since 2001 the level of synchronization of large price movements across assets has significantly increased. We find that only a minor fraction of these systemic events can be connected with the release of pre-announced macroeconomic news. Finally, the larger is the multiplicity of the event—i.e. how many assets have swung together—the larger is the probability of a new event occurring in the near future, as well as its multiplicity. To reproduce these facts, due to the self- and cross-exciting nature of the event dynamics, we propose an approach based on Hawkes processes. For each event, we directly model the multiplicity as a multivariate point process, neglecting the identity of the specific assets. This allows us to introduce a parsimonious parametrization of the kernel of the process and to achieve a reliable description of the dynamics of large price movements for a high-dimensional portfolio.