In this paper, we aim to establish some stylized facts about the relative contributions of the job-finding and separation rates to unemployment dynamics depending on the nature of structural shocks. The shocks in our Bayesian Structural VAR model are identified using a sign-restriction approach, and capture shifts in the three conditions determining labor-market equilibrium in matching models: the Beveridge curve, and the job-creation and job-destruction conditions. Using both US and French data, we identify an aggregate shock to match profitability (the aggregate-profitability shock), a shock specific to existing jobs (the reallocation shock) and a shock to the efficiency of the matching process (the matching-efficiency shock). We find that the relative contributions of the job finding and separation rates are notably different across shocks in both countries, but are similar across countries for each type of shock. Labor market dynamics appear to be Transatlantic despite well-known institutional differences. However, it must be emphasized that the reaction of the labor market variables to the matching-efficiency shock seems more muted in the US than in France.