Giant planets can interact with multiple and chemically diverse environments in protoplanetary discs while they form and migrate to their final orbits. The way this interaction affects the accretion of gas and solids shapes the chemical composition of the planets and of their atmospheres. Here we investigate the effects of different chemical structures of the host protoplanetary disc on the planetary composition. We consider both scenarios of molecular (inheritance from the pre-stellar cloud) and atomic (complete chemical reset) initial abundances in the disc. We focus on four elemental tracers of different volatility: C, O, N, and S. We explore the entire extension of possible formation regions suggested by observations by coupling the disc chemical scenarios with N-body simulations of forming and migrating giant planets. The planet formation process produces giant planets with chemical compositions significantly deviating from that of the host disc. We find that the C/N, N/O, and S/N ratios follow monotonic trends with the extent of migration. The C/O ratio shows a more complex behaviour, dependent on the planet accretion history and on the chemical structure of the formation environment. The comparison between S/N* and C/N* (where * indicates normalisation to the stellar value), constrains the relative contribution of gas and solids to the total metallicity. Giant planets whose metallicity is dominated by the contribution of the gas are characterised by N/O* > C/O* > C/N* and allow to constrain the disc chemical scenario. When the planetary metallicity is instead dominated by the contribution of the solids we find that C/N* > C/O* > N/O*.