We investigate the nature of the deconfinement transitions in three-dimensional lattice Abelian Higgs models, in which a complex scalar field of integer charge Q≥2 is minimally coupled with a compact U(1) gauge field. Their phase diagram presents two phases separated by a transition line where static charges q, with q<Q, deconfine. We argue that these deconfinement transitions belong to the same universality class as transitions in generic three-dimensional Z_{Q} gauge models. In particular, they are Ising-like for Q=2, of first order for Q=3, and belong to the three-dimensional gauge XY universality class for Q≥4. This general scenario is supported by numerical finite-size scaling analyses of the energy cumulants for Q=2, Q=4, and Q=6.