This work focuses on the solution of a constrained multiobjective optimization problem, with both nonlinear inequality constraints and bound constraints. We assume that the vector of the objective functions and the constraints are Lipschitz continuous. We issue the equivalence between the original constrained multiobjective problem, and a multiobjective problem with simple bounds, by means of an exact penalty function approach. We study the Pareto-Clarke stationary points of the multiobjective problem with bound constraints, and state their correspondence with the Pareto-Clarke stationary points of the original constrained multiobjective problem. We propose a line search based derivative free framework to issue the latter correspondence. We also report some numerical results proving the effectiveness of the proposed approach.