  We present a real-time diagrammatic formalism to study adiabatic pumping in
chains of tunnel-coupled metallic islands. This approach is based on an
expansion to linear order in the frequency of the time-dependent parameters and
on a systematic perturbation expansion in the tunnel-coupling strength. We
apply our formalism to a single-island and a double-island system. In the
single-island setup, we find that the first-order contribution in the
tunnel-coupling strength is purely due to the renormalization of the
charging-energy gap. In the double-island system, we investigate the transition
between weak and strong pumping.
