The dynamics of an x-ray-ionized two-component core-shell nanosystem is probed using doped helium (He) nanodroplets. First, a soft x-ray pump pulse selectively inner-shell ionizes the core cluster formed of heavier rare-gas atoms, causing electron migration from the He shell to the highly charged core. This ignites a He nanoplasma which is then driven by an intense near-infrared probe pulse. The ultrafast charge redistribution, evidenced by the rise of He$^+$ and He$^{2+}$ ion yields from the nanoplasma within $<70$ fs, leads to strong damping of the core cluster expansion. Thus, He droplets act as efficient tampers that reduce the radiation damage of embedded nanostructures, a property that could be exploited for improving coherent diffraction images.
