  In this paper we describe a general method to derive formulas relating the
gap probability of some classical determinantal random point process (Airy,
Pearcey and Hermite) with the gap probability of the processes related to the
same kernels with "wanderers", "inliers" and "outliers". In this way, we
generalize the Painlev\'e-like formula found by Baik for the Baik-Ben
Arous-P\'ech\'e distribution to many different cases, both in the one and
multi-time case. The method is not ad-hoc and relies upon the notion of
discrete Schlesinger transformations for Riemann-Hilbert problems.
