When using bifunctional core@shell catalysts, the stability of both the shell and core–shell interface is crucial for catalytic applications. In the present study, we elucidate the stability of a CuO/ZnO/Al2O3@ZSM-5 core@shell material, used for one-stage synthesis of dimethyl ether from synthesis gas. The catalyst stability was studied in a hierarchical manner by complementary environmental transmission electron microscopy (ETEM), scanning electron microscopy (SEM) and in situ hard X-ray ptychography with a specially designed in situ cell. Both reductive activation and reoxidation were applied. The core–shell interface was found to be stable during reducing and oxidizing treatment at 250°C as observed by ETEM and in situ X-ray ptychography, although strong changes occurred in the core on a 10 nm scale due to the reduction of copper oxide to metallic copper particles. At 350°C, in situ X-ray ptychography indicated the occurrence of structural changes also on the µm scale, i.e. the core material and parts of the shell undergo restructuring. Nevertheless, the crucial core–shell interface required for full bifunctionality appeared to remain stable. This study demonstrates the potential of these correlative in situ microscopy techniques for hierarchically designed catalysts.