Context. Most of the studies on the determination of the chemical composition of metal-poor stars have been focused on the search of the most pristine stars, searching for the imprints of the ejecta of the first supernovae. Apart from the rare and very interesting r-enriched stars, few elements are measurable in the very metal-poor stars. On the other hand, a lot of work has been done also on the thin-disc and thick-disc abundance ratios in a metallicity range from [Fe/H]> −1.5 dex to solar. In the available literature, the intermediate metal-poor stars (−2.5<[Fe/H]< −1.5) have been frequently overlooked. The MINCE (Measuring at Intermediate metallicity Neutron-Capture Elements) project aims to gather the abundances of neutron-capture elements but also of light elements and iron peak elements in a large sample of giant stars in this metallicity range. The missing information has consequences for the precise study of the chemical enrichment of our Galaxy in particular for what concerns neutron-capture elements and it will be only partially covered by future multi object spectroscopic surveys such as WEAVE and 4MOST. Aims. The aim of this work is to study the chemical evolution of galactic sub-components recently identified (i.e. Gaia Sausage Enceladus (GSE), Sequoia). Methods. We used high signal-to-noise ratios, high-resolution spectra and standard 1D LTE spectrum synthesis to determine the detailed abundances. Results. We could determine the abundances for up to 10 neutron-capture elements (Sr, Y, Zr, Ba, La, Ce, Pr, Nd, Sm and Eu) in 33 stars. The general trends of abundance ratios [n-capture element/Fe] versus [Fe/H] are in agreement with the results found in the literature. When our sample is divided in sub-groups depending on their kinematics, we found that the run of [Sr/Ba] versus [Ba/H] for the stars belonging to the GSE accretion event shows a tight anti-correlation. The results for the Sequoia stars, although based on a very limited sample, shows a [Sr/Ba] systematically higher than the [Sr/Ba] found in the GSE stars at a given [Ba/H] hinting at a different nucleosynthetic history. Stochastic chemical evolution models have been computed to understand the evolution of the GSE chemical composition of Sr and Ba. The first conclusions are that the GSE chemical evolution is similar to the evolution of a dwarf galaxy with galactic winds and inefficient star formation. Conclusions. Detailed abundances of neutron-capture elements have been measured in high-resolution, high signal-to-noise spectra of intermediate metal-poor stars, the metallicity range covered by the MINCE project. These abundances have been compared to detailed stochastic models of galactic chemical evolution.