We present the analysis of physical conditions, chemical composition, and kinematic properties of two bow shocks - HH 529 II and HH 529 III - of the fully photoionized Herbig-Haro object HH 529 in the Orion Nebula. The data were obtained with the Ultraviolet and Visual Echelle Spectrograph at the 8.2m Very Large Telescope and 20 yr of Hubble Space Telescope imaging. We separate the emission of the high-velocity components of HH 529 II and III from the nebular one, determining ne and Te in all components through multiple diagnostics, including some based on recombination lines (RLs). We derive ionic abundances of several ions, based on collisionally excited lines and RLs. We find a good agreement between the predictions of the temperature fluctuation paradigm (t2) and the abundance discrepancy factor (ADF) in the main emission of the Orion Nebula. However, t2 cannot account for the higher ADF found in HH 529 II and III. We estimate 6 per cent of Fe in the gas phase of the Orion Nebula, while this value increases to 14 per cent in HH 529 II and between 10 and 25 per cent in HH 529 III. We find that such increase is probably due to the destruction of dust grains in the bow shocks. We find an overabundance of C, O, Ne, S, Cl, and Ar of about 0.1 dex in HH 529 II and III that might be related to the inclusion of H-deficient material from the source of the HH 529 flow. We determine the proper motions of HH 529 finding multiple discrete features. We estimate a flow angle with respect to the sky plane of 58° ± 4° for HH 529.