We report on radiotracer diffusion measurements in metallic bulk-glass-forming Pd-Cu-Ni-P alloys. The Pd-Cu-Ni-P system, with its high stability against crystallization, allows diffusion measurements from the glassy state to the equilibrium melt for the first time. Serial sectioning was performed by grinding and ion-beam sputtering. The time and temperature as well as mass dependence, expressed in terms of the isotope effect E, of codiffusion were investigated. In the glassy state as well as in the deeply supercooled state below the critical temperature Tc, where the mode-coupling theory predicts a freezing-in of liquidlike motion, the measured very small isotope effects indicated a highly collective hopping mechanism. Below Tc, the temperature dependence showed Arrhenius-type behavior. Above Tc, the onset of liquidlike motion was evidenced by a gradual drop of the effective activation energy, resulting from the decay of hopping barriers, and by the validity of the Stokes-Einstein equation, which was found to break down below Tc. This strongly supports the mode-coupling scenario. Isotope effect measurements, which have never been carried out near Tc in any material, showed atomic transport up to the equilibrium melt to be far away from the hydrodynamic regime of uncorrelated binary collisions. The latter appears to be a prerequisite of excellent glass-forming abilities.