  All Painlev\'e equations can be written as a time-dependent Hamiltonian
system, and as such they admit a natural generalization to the case of several
particles with an interaction of Calogero type (rational, trigonometric or
elliptic). Recently, these systems of interacting particles have been proved to
be relevant in the study of $\beta$-models. An almost two decade old open
question by Takasaki asks whether these multi-particle systems can be
understood as isomonodromic equations, thus extending the Painlev\'e
correspondence. In this paper we answer in the affirmative by displaying
explicitly suitable isomonodromic Lax pair formulations. As an application of
the isomonodromic representation we provide a construction based on discrete
Schlesinger transforms, to produce solutions for these systems for special
values of the coupling constants, starting from uncoupled ones; the method is
illustrated for the case of the second Painlev\'e equation.
