This work investigates a nonsmooth multiobjective semi-infinite programming problem with switching constraints \(({\mathcal {M}}{\mathcal {S}}{\mathcal {P}}{\mathcal {S}}{\mathcal {C}})\) . Using convexificators and under the nonsmooth Abadie constraint qualification, we derive necessary optimality conditions in the form of \(\textbf{M}\) -stationary. Furthermore, under generalized convexity conditions expressed through the properties of convexificators, we establish sufficient conditions for optimality. For the original \({\mathcal {M}}{\mathcal {S}}{\mathcal {P}}{\mathcal {S}}{\mathcal {C}}\) problem, we formulate a Mond-Weir-type dual model and prove both weak and strong duality results between the primal and dual problems. Finally, numerical examples are provided to demonstrate the applicability of the theoretical findings to \({\mathcal {M}}{\mathcal {S}}{\mathcal {P}}{\mathcal {S}}{\mathcal {C}}\) .