Darboux transformations and random point processes

40 pages, 1 figure (only!), ver2, grammatical correctionsInternational audienceIn this paper we describe a general method to derive formulas relating the gap probability of some classical determinantal random point processes (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 Painlevé-like formula found by Baik for the Baik-Ben Arous-Péché 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 Schlesinger transformations for Riemann-Hilbert problems