Since the discovery of bulk metallic glasses there has been considerable research effort on these systems, in particular with respect to mass transport. Now the undercooled melt between the melting temperature and the caloric glass transition temperature, which has not been accessible before due to the rapid onset of crystallization, can be investigated and theories can be tested. Here we report on radiotracer diffusion measurements in metallic bulk-glass-forming Pd-Cu-Ni-P alloys. Serial sectioning was performed by grinding and ion-beam sputtering. The time, temperature as well as the mass dependence, expressed in terms of the isotope effect E, of Co-diffusion were investigated. The Co isotope effect measurements, which have never been carried out near Tc in any material, show atomic transport up to the equilibrium melt to be far away from the hydrodynamic regime of uncorrelated binary collisions. 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 liquid-like motion, the experimentally determined very small isotope effects indicate a highly collective hopping mechanism involving some ten atoms. Below Tc the temperature dependence shows Arrhenius-type behavior with an effective activation enthalpy of 3.2 eV. Above Tc the onset of liquid-like motion is evidenced by a gradual drop of the effective activation energy and by the validity of the Stokes-Einstein equation, which is found to break down below Tc. Although having strong covalent bonding tendencies, Phosphorous diffusion is only slightly slower than Co diffusion, indicating that it does not determine the overall viscosity below Tc. The Stokes-Einstein equation is presently tested for other constituents of the alloy.