  The variation of the expansion rate of the Universe with time produces an
evolution in the cosmological redshift of distant sources (for example quasar
Lyman-$\alpha$ absorption lines), that might be directly observed by future
ultra stable, high-resolution spectrographs (such as CODEX) coupled to
extremely large telescopes (such as European Southern Observatory's Extremely
Large Telescope, ELT). This would open a new window to explore the physical
mechanism responsible for the current acceleration of the Universe. We
investigate the evolution of cosmological redshift from a variety of dark
energy models, and compare it with simulated data. We perform a Fisher matrix
analysis and discuss the prospects for constraining the parameters of these
models and for discriminating among competing candidates. We find that, because
of parameter degeneracies, and of the inherent technical difficulties involved
in this kind of observations, the uncertainties on parameter reconstruction can
be rather large unless strong external priors are assumed. However, the method
could be a valuable complementary cosmological tool, and give important
insights on the dynamics of dark energy, not obtainable using other probes.
