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 Observatorys 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.
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