Context. Stellar jets are an essential ingredient of the star formation process and a wealth of information can be derived from their characteristic emission-line spectra. Aims. We investigate the physical structure and dust reprocessing in the shocks along the beam of a number of classical Herbig-Haro (HH) jets in the Orion and Lupus molecular clouds (HH 111, HH 1/2, HH 83, HH 24 M/A/E/C, and Sz68). Parameters describing plasma conditions, as well as dust content, are derived as a function of distance from the source and, for HH 111, of gas velocity. Methods. Spectral diagnostic techniques are applied to obtain the jet physical conditions (the electron and total density, n e and n H , the ionisation fraction, x e , and the temperature, T e ) from the ratios of selected forbidden lines. The presence of dust grains is investigated by estimating the gas-phase abundance of calcium with respect to its solar value. Results. We find the electron density varies between 0.05-4 × 10 3 cm −3 , the ionisation fraction x e from 0.01-0.7, the temperature ranges between 0.6-3 × 10 4 K, and the hydrogen density between 0.01-6 × 10 4 cm −3 . Interestingly, in the HH 111 jet, n e , x e , and T e peak in the high velocity interval (HVI) of the strongest working surfaces, confirming a prediction from shocks models. Calcium turns out to be depleted with respect to its solar value, but its gas-phase abundance is higher than estimates for the interstellar medium in Orion. The depletion is high (up to 80%) along the low-excited jets, while low or no depletion is measured in those jets which show higher excitation conditions. Moreover, for HH 111 the depletion is lower in the HVI of the fastest shock. Conclusions. Our results confirm the shock structure predicted by models and indicate that shocks occurring along jets, and presumably those present in the launch zone, only partially destroy dust grains and that the efficiency of dust reprocessing strongly depends on shock velocity. However, the high Ca gas-phase abundance estimated in some of the knots, is not well understood in terms of existing models of dust reprocessing in shocks, and indicates that the dust must have been partially reprocessed in the region where the flow originates.