In this paper we investigate some aspects of stochastic local search such as pressure toward and along the set of interest within parameter dependent multi-objective optimization problems. The discussions and initial computations indicate that the problem to compute an approximation of the entire solution set of such a problem via stochastic search algorithms is well-conditioned. The new insights may be helpful for the design of novel stochastic search algorithms such as specialized evolutionary approaches. The discussion in particular indicates that it might be beneficial to integrate the set of external parameters directly into the search instead of computing projections of the solution sets separately by fixing the value of the external parameter.