Context. Atomic diffusion and mixing processes in stellar interiors influence the structure and the surface composition of stars. Some of these processes cannot yet be modelled from the first principles, and they require calibrations. This limits their applicability in stellar models used for studies of stellar populations and Galactic evolution.Aims. Our main goal is to put constraints on the stellar structure and evolution models using new refined measurements of the chemical composition in stars of a Galactic open cluster.Methods. We used medium-resolution, 19 200 <= R <= 21 500, optical spectra of stars in the open cluster NGC 2420 obtained within the Gaia-ESO survey. The sample covers all evolutionary stages from the main sequence to the red giant branch. Stellar parameters were derived using a combined Bayesian analysis of spectra, 2MASS photometry, and astrometric data from Gaia DR2. The abundances of Mg, Ca, Fe, and Li were determined from non-local thermodynamic equilibrium (NLTE) synthetic spectra, which were computed using one-dimensional (1D) and averaged three-dimensional (3D) model atmospheres. We compare our results with a grid of Code d'Evolution Stellaire Adaptatif et Modulaire (CESTAM) stellar evolution models, which include atomic diffusion, turbulent, and rotational mixing.Results. We find prominent evolutionary trends in the abundances of Fe, Ca, Mg, and Li with the mass of the stars in the cluster. Furthermore, Fe, Mg, and Ca show a depletion at the cluster turn-off, but the abundances gradually increase and flatten near the base of the red giant branch. The abundance trend for Li displays a signature of rotational mixing on the main sequence and abrupt depletion on the sub-giant branch, which is caused by advection of Li-poor material to the surface. The analysis of abundances combined with the CESTAM model predictions allows us to place limits on the parameter space of the models and to constrain the zone in the stellar interior, where turbulent mixing takes place.