We carried out a comprehensive analysis of the chemical properties of the interstellar medium (ISM) and the stellar population (SP) of current normal galaxies and their progenitors in a hierarchical clustering scenario. We compared the results with observations of damped Lyman- α systems (DLAs) under the hypothesis that, at least, part of the observed DLAs could originate in the building blocks of present-day normal galaxies. We used a hydrodynamical cosmological code which includes star formation and chemical enrichment. Galaxy-like objects are identified at z = 0 and then followed back in time. Random lines of sight (LOS) are drawn through these structures in order to mimic damped Lyman-α systems. We then analysed the chemical properties of the ISM and SP along the LOS. We found that the progenitors of current galaxies in the field with mean L < 0.5L* and virial circular velocity of 100-250 km s-1 could be the associated DLA galaxies. For these systems we detected a trend for (L/L*) to increase with redshift. We found moderate metallicity evolution for [Zn/H], [Fe/H] and [Si/H]. However, when we applied the observational filter suggested by Boissé et al. (1998) in order to restrict the sample to the observed limits in densities and metallicities, we found mild evolution consistent with observational results that include dust corrections. [Si/Fe] and [S/Fe] show weak α-enhancement in agreement with observations corrected by dust depletion. We found α/Fe in the ISM and SP to have more homogeneous abundances than [Fe/H] and [Zn/H]. In our models, the global metallicity evolution is driven by the high metallicity and high column density simulated DLAs, which have low impact parameters (b < 5 kpc), and SPs with more than 108 M⊙. Our results suggest that geometrical effects could be the mechanism responsible for the non-detectability of high-metallicity and high-column-density DLAs. We found sub-DLAs to map preferentially the outskirts of the simulated DLA galaxies. Hence, they can contribute to the study of the metallicity of the galactic structure as a function of redshift. An analysis of the metallicity content of the ISMs and SPs of the galaxy-like objects as a function of redshift shows the formation of a central stellar mass concentration with nearly solar metallicity at all redshifts while stars in the outer parts of these objects have lower metallicities. The gas content becomes enriched progressively with redshift and at all radii. The abundance properties of the galaxy-like objects and the simulated DLAs are the results of the contribution of type la and II Supernovae and gas infall from the dark matter haloes with a timing settled by their particular evolution history in a hierarchical clustering scenario. Our results suggest that the mild evolution detected in the observations could arise from a conspiracy of all of these processes.