  A time-dependent approach is used to explore inelastic effects during
electron transport through few-level systems. We study a tight-binding chain
with one and two sites connected to vibrations. This simple but transparent
model gives insight about inelastic effects, their meaning and the
approximations currently used to treat them. Our time-dependent approach allows
us to trace back the time sequence of vibrational excitation and electronic
interference, the ibrationally introduced time delay and the electronic phase
shift. We explore a full range of parameters going from weak to strong
electron-vibration coupling, from tunneling to contact, from one-vibration
description to the need of including all vibrations for a correct description
of inelastic effects in transport. We explore the validity of single-site
resonant models as well as its extension to more sites via molecular orbitals
and the conditions under which multi-orbital, multi-vibrational descriptions
cannot be simplified. We explain the physical meaning of the spectral features
in the second derivative of the electron current with respect to the bias
voltage. This permits us to nuance the meaning of the energy value of dips and
peaks. Finally, we show that finite-band effects lead to electron
back-scattering off the molecular vibrations in the regime of high-conductance,
although the drop in conductance at the vibrational threshold is rather due to
the rapid variation of the vibronic density of states.
