  We investigate equilibrium and non-equilibrium charge-transfer processes by
performing high-resolution transport spectroscopy. Using electrostatically
defined quantum dots for energy-selective emission and detection, we achieved
unprecedented spectral resolution and a high degree of tunability of relevant
experimental parameters. Most importantly, we observe that the spectral width
of elastically transferred electrons can be substantially smaller than the
linewidth of a thermally broadened Coulomb peak. This finding indicates that
the charge-transfer process is fast compared to the electron--phonon
interaction time. By drawing an analogy to double quantum dots, we argue that
the spectral width of the elastic resonance is determined by the lifetime
broadening $h\it{\Gamma}$ of the emitter and detector states. Good agreement
with the model is found also in an experiment in which the charge transfer is
in the regime $h\it{\Gamma}\gg k_{\rm{B}}T$. By performing spectroscopy below
the Fermi energy, we furthermore observe elastic and inelastic transfer of
holes.
