Context. Young open clusters (ages of less than 200 Myr) have been observed to exhibit several peculiarities in their chemical compositions. These anomalies include a slightly sub-solar iron content, super-solar abundances of some atomic species (e.g. ionised chromium), and atypical enhancements of [Ba/Fe], with values up to ~0.7 dex. Regarding the behaviour of the other s-process elements like yttrium, zirconium, lanthanum, and cerium, there is general disagreement in the literature: some authors claim that they follow the same trend as barium, while others find solar abundances at all ages. Aims. In this work we expand upon our previous analysis of a sample of five young open clusters (IC 2391, IC 2602, IC 4665, NGC 2516, and NGC 2547) and one star-forming region (NGC 2264), with the aim of determining abundances of different neutron-capture elements, mainly Cu I, Sr I, Sr II, Y II, Zr II, Ba II, La II, and Ce II. For NGC 2264 and NGC 2547 we present the measurements of these elements for the first time. Methods. We analysed high-resolution, high signal-to-noise spectra of 23 solar-type stars observed within the Gaia-ESO survey. After a careful selection, we derived abundances of isolated and clean lines via spectral synthesis computations and in a strictly differential way with respect to the Sun. Results. We find that our clusters have solar [Cu/Fe] within the uncertainties, while we confirm that [Ba/Fe] is super-solar, with values ranging from +0.22 to +0.64 dex. Our analysis also points to a mild enhancement of Y, with [Y/Fe] ratios covering values between 0 and +0.3 dex. For the other s-process elements we find that [X/Fe] ratios are solar at all ages. Conclusions. It is not possible to reconcile the anomalous behaviour of Ba and Y at young ages with standard stellar yields and Galactic chemical evolution model predictions. We explore different possible scenarios related to the behaviour of spectral lines, from the dependence on the different ionisation stages and the sensitivity to the presence of magnetic fields (through the Landé factor) to the first ionisation potential effect. We also investigate the possibility that they may arise from alterations of the structure of the stellar photosphere due to the increased levels of stellar activity that affect the spectral line formation, and consequently the derived abundances. These effects seem to be stronger in stars at ages of less than ~ 100 Myr. However, we are still unable to explain these enhancements, and the Ba puzzle remains unsolved. With the present study we suggest that other elements, for example Sr, Zr, La, and Ce, might be more reliable tracer of the s-process at young ages, and we strongly encourage further critical observations.