  The local conservation of a physical quantity whose distribution changes with
time is mathematically described by the continuity equation. The corresponding
time parameter, however, is defined with respect to an idealized classical
clock. We consider what happens when this classical time is replaced by a
non-relativistic quantum-mechanical description of the clock. From the
clock-dependent Schr\"odinger equation (as analogue of the time-dependent
Schr\"odinger equation) we derive a continuity equation, where, instead of a
time-derivative, an operator occurs that depends on the flux (probability
current) density of the clock. This clock-dependent continuity equation can be
used to analyze the dynamics of a quantum system and to study degrees of
freedom that may be used as internal clocks for an approximate description of
the dynamics of the remaining degrees of freedom. As an illustration, we study
a simple model for coupled electron-nuclear dynamics and interpret the nuclei
as quantum clock for the electronic motion. We find that whenever the
Born-Oppenheimer approximation is valid, the continuity equation shows that the
nuclei are the only relevant clock for the electrons.
