Context. Studies of the dust extinction in the Galaxy have found that the large majority of sight lines analyzed obey a simple relation depending on one parameter, the total-to-selective extinction coefficient, R V . Different values of R V are able to match the whole extinction curve in different environments, thus characterize normal extinction curves. However anomalous curves i.e, curves which deviate strongly from this simple behavior have been observed in the Galaxy as well as external galaxies. Aims. More than sixty curves with large ultraviolet deviations from their best-fit one-parameter curve are analyzed. The extinction curves are fitted with dust models to shed light on the properties of the grains along selected lines of sight, the processes affecting them, and their relations with the environmental characteristics. Methods. The extinction curve models are developed using the latest prescriptions for grain-size distributions capable of describing one-parameter curves for R V values from 3.1 to 5.5. These models, extended here down to R V = 2.0, allow us to compare the resulting properties of our deviating curves with those of normal curves in a self-consistent framework, and thus to recover the relative trends overcoming the modeling uncertainties.Results. In addition to 20 anomalous curves extracted from the same sample for study in a previous paper and revised here to account for the latest updates, these curves represent the largest, homogeneous sample of anomalous curves studied so far with dust models. Our results show that the ultraviolet deviations are driven by a larger amount of small grains than predicted for lines of sight where extinction depends on only one parameter. Moreover, the dust-to-gas ratios of anomalous curves are lower than the same values for no deviating lines of sight. Conclusions. Both shocks and grain-grain collisions should destroy dust grains, hence reduce the amount of the dust trapped in the grains, modify the size distribution of the dust, and increase the small-to-large grain size ratio. Therefore, the extinction properties derived should correspond to sight lines where shocks and high velocity flows perturb the physical state of the interstellar medium and leave their signature on the dust properties.