  We find an exact expression for the current ($I$) that flows via a tagged
bond from a site ("dot") whose potential ($u$) is varied in time. We show that
the analysis reduces to that of calculating time dependent probabilities, as in
the stochastic formulation, but with splitting (branching) ratios that are not
bounded within $[0,1]$. Accordingly our result can be regarded as a
multiple-path version of the continuity equation. It generalizes results that
have been obtained from adiabatic transport theory in the context of quantum
"pumping" and "stirring". Our approach allows to address the adiabatic regime,
as well as the Slow and Fast non-adiabatic regimes, on equal footing. We
emphasize aspects that go beyond the familiar picture of sequential
Landau-Zener crossings, taking into account Wigner-type mixing of the energy
levels.
