Asteroid dynamical families are supposed to be formed from the collisional disruption of parent bodies. As a consequence, the investigation of the surface properties of small and large family members may give some hints on the nature of the dynamical group, the internal composition of the parent body, and the role played by space weathering processes in modifying the spectral behavior of the members' surfaces. In this work we present visible near-infrared observations of 24 Jupiter Trojans belonging to seven dynamical families of both the L4 and L5 swarms. The most important characteristics we found is the uniformity of the Trojans population. All the investigated Trojans have featureless spectra and a spectral behavior typical of the primitive P and D taxonomic classes. In particular, no signatures of water ice have been found on the spectra of these primordial bodies. From our investigation, the L4 and L5 clouds appear to be compositionally indistinguishable. Tentative models of the surface composition, based on the Hapke theory, are presented and discussed.