We have studied the existence of self-dual effective compact and true compacton configurations in Abelian Higgs models with generalized dynamics. We have named an effective compact solution the one whose profile behavior is very similar to the one of a compacton structure but still preserves a tail in its asymptotic decay. In particular, we have investigated the electrically neutral configurations of the Maxwell-Higgs and Born-Infeld-Higgs models and the electrically charged ones of the Chern-Simons-Higgs and Maxwell-Chern-Simons-Higgs models. The generalization of the kinetic terms is performed by means of dielectric functions in gauge and Higgs sectors. The implementation of the BPS formalism without the need to use a specific Ansatz has led us to the explicit determination for the dielectric function associated with the Higgs sector to be proportional to φ2-2, >1. Consequently, the followed procedure allows us to determine explicitly new families of self-dual potential for every model. We have also observed that, for sufficiently large values of, every model supports effective compact vortices. The true compacton solutions arising for =∞ are analytical. Therefore, these new self-dual structures enhance the space of BPS solutions of the Abelian Higgs models and they probably will imply interesting applications in physics and mathematics.