We provide evidence for this conclusion: given a finite Galois cover of group, almost all (in a density sense) realizations of over do not occur as specializations of. We show that this holds if the number of branch points of is sufficiently large, under the abc-conjecture and, possibly, the lower bound predicted by the Malle conjecture for the number of Galois extensions of of given group and bounded discriminant. This widely extends a result of Granville on the lack of -rational points on quadratic twists of hyperelliptic curves over with large genus, under the abc-conjecture (a diophantine reformulation of the case of our result). As a further evidence, we exhibit a few finite groups for which the above conclusion holds unconditionally for almost all covers of of group. We also introduce a local-global principle for specializations of Galois covers and show that it often fails if has abelian Galois group and sufficiently many branch points, under the abc-conjecture. On the one hand, such a local-global conclusion underscores the 'smallness' of the specialization set of a Galois cover of. On the other hand, it allows to generate conditionally 'many' curves over failing the Hasse principle, thus generalizing a recent result of Clark and Watson devoted to the hyperelliptic case.