We analyze how the transmission of international inflation spillovers depends on the nature of the underlying shocks that drive inflation abroad. We find evidence for substantial heterogeneity in the magnitude of spillovers to domestic inflation related to the fundamental source of international price fluctuations and the corresponding monetary policy reactions. Indeed, it turns out that the relative conduct of monetary policy varies depending on the source of these price fluctuations, and so does the role of the exchange rate as a shock absorber. We show this by looking at international inflation spillovers to Switzerland through the lenses of a Bayesian structural dynamic factor model relating a large set of disaggregated prices to key macroeconomic factors. Being a small open economy with an independent monetary policy, Switzerland is a particularly suitable subject for studying the role of monetary policy in the transmission of foreign shocks. However, our results more broadly indicate that inflation spillovers need to be analyzed in a framework allowing for different transmission channels.