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We consider a variant of the charge-Q compact Abelian-Higgs model, in which an Nf-dimensional complex vector is coupled with an Abelian Z_q gauge field. For Nf=2 and Q=1 we observe several transition lines that belong to the O(4), O(3), and O(2) vector universality classes, depending on the symmetry breaking pattern at the transition. The universality class is independent of $q$ as long as q>=3. The universality class of the transition is uniquely determined by the behavior of the scalar fields; gauge fields do not play any role. We also investigate the system for Nf=15 and Q=2. In the presence of U(1) gauge fields, the system undergoes transitions associated with charged fixed points of the Abelian-Higgs field theory. These continuous transitions turn into first-order ones when the U(1) gauge fields are replaced by the discrete Z_q fields: in the present compact model charged transitions appear to be very sensitive to the nature of the gauge fields, Comment: 25 pages, 15 figures