We investigate the complexity of axiom pinpointing for different members of the DL-Lite family of Description Logics. More precisely, we consider the problem of enumerating all minimal subsets of a given DL-Lite knowledge base that have a given consequence. We show that for the DL-Lite^{[Hscr ]}_core, DL-Lite^{[Hscr ]}_krom and DL-Lite^{[Hscr ][Nscr ]}_horn fragments such minimal subsets are efficiently enumerable with polynomial delay, but for the DL-Lite_bool fragment they cannot be enumerated in output polynomial time unless P = NP. We also show that interestingly, for the DL-Lite^{[Hscr ][Nscr ]}_horn fragment such minimal sets can be enumerated in reverse lexicographic order with polynomial delay, but it is not possible in the forward lexicographic order since computing the first one is already coNP-hard.