We present an approach for initiating and tracing ultrafast electron dynamics in core-excited atoms, molecules, and solids. The approach is based on stimulated resonant inelastic x-ray scattering induced by a single, chirped, broadband XUV or x-ray pulse. A first interaction with this pulse prepares a core-excited state wave packet by resonant core-excitation. A second interaction with the pulse at a later time induces the transition to valence-excited states, which is associated with stimulated emission. The preparation of the core-excited wave packet and the transition from the core-excited states to the valence-excited states occur at distinct chirp-dependent times. As a consequence, the stimulated emission carries information about the time evolution of the core-excited state wave packet.