Context. Several works have found an increase of the abundances of the s-process neutron-capture elements in the youngest Galactic stellar populations. These trends provide important constraints on stellar and Galactic evolution and they need to be confirmed with large and statistically significant samples of stars spanning wide age and distance intervals. Aims: We aim to trace the abundance patterns and the time evolution of five s-process elements - two belonging to the first peak, Y and Zr, and three belonging to the second peak, Ba, La, and Ce - using the Gaia-ESO IDR5 results for open clusters and disc stars. Methods: From the UVES spectra of cluster member stars, we determined the average composition of clusters with ages >0.1 Gyr. We derived statistical ages and distances of field stars, and we separated them into thin and thick disc populations. We studied the time-evolution and dependence on metallicity of abundance ratios using open clusters and field stars whose parameters and abundances were derived in a homogeneous way. Results: Using our large and homogeneous sample of open clusters, thin and thick disc stars, spanning an age range larger than 10 Gyr, we confirm an increase towards young ages of s-process abundances in the solar neighbourhood. These trends are well defined for open clusters and stars located nearby the solar position and they may be explained by a late enrichment due to significant contribution to the production of these elements from long-living low-mass stars. At the same time, we find a strong dependence of the s-process abundance ratios on the Galactocentric distance and on the metallicity of the clusters and field stars. Conclusions: Our results, derived from the largest and most homogeneous sample of s-process abundances in the literature, confirm the growth with decreasing stellar ages of the s-process abundances in both field and open cluster stars. At the same time, taking advantage of the abundances of open clusters located in a wide Galactocentric range, these results offer a new perspective on the dependence of the s-process evolution on the metallicity and star formation history, pointing to different behaviours at various Galactocentric distances. Based on observations collected with the FLAMES instrument at VLT/UT2 telescope (Paranal Observatory, ESO, Chile), for the Gaia-ESO Large Public Spectroscopic Survey (188.B-3002, 193.B-0936).