Through approximate solutions to the equation of motion for density matrices coupled with the first-principles Hamiltonian, we derive a rate equation describing time evolutions of core-valence photoexcitation and collisional relaxation in metals irradiated by VUV/x-ray free-electron lasers. We discuss how to treat many-body effects such as energy-level renormalization, absorption-edge shift and electron screening caused by interactions among electrons and core holes.