<p>We first extend the product Chern-Simons theory to develop several mixed <i>U</i> (1) × <i>U</i> (1) models where one gauge field is governed by a Chern-Simons term and the other by a Maxwell or Born-Infeld term. We show that, by choosing suitable potentials, the energy functional admits a topological lower bound saturated by first-order self-dual equations. The resulting dyonic systems can be divided into vortex-vortex and vortex-antivortex configurations, and the coexistence of vortices and antivortices in the latter extends the vortex-only result known in the product Chern-Simons model. On a doubly periodic domain, we establish Bradlow-type bounds with distinct physical implications: for vortex-only systems, the vortex numbers stay below these bounds and cannot be arbitrarily large; for vortex-antivortex systems, the bounds are imposed on the difference between the vortex and antivortex numbers, while the individual numbers are arbitrary. This distinction results in a bounded energy spectrum for the former and an unbounded energy spectrum for the latter. Furthermore, we extend the above product vortex-only and vortex-antivortex models to the general case of <i>U</i> (1)^<i>n</i>, where the Lagrangian includes <i>m</i> Chern-Simons terms, <i>t</i> − <i>m</i> Maxwell terms and <i>n</i> − <i>t</i> Born-Infeld terms. We then investigate the Bogomol’nyi structure following the same method as in the <i>U</i> (1) × <i>U</i> (1) case.</p>