  A simultaneous description of the dynamics of multiple particles requires a
configuration space approach with an external time parameter. This is in stark
contrast with the relativistic paradigm, where time is but a coordinate chosen
by an observer. Here we show, however, that the two attitudes toward modelling
$N$-particle dynamics can be conciliated within a generally covariant
framework. To this end we construct an '$N$-particle configuration spacetime'
$\mathcal{M}_{\scriptscriptstyle (N)}$, starting from a globally hyperbolic
spacetime $\mathcal{M}$ with a chosen smooth splitting into time and space
components. The dynamics of multi-particle systems is modelled at the level of
Borel probability measures over $\mathcal{M}_{\scriptscriptstyle (N)}$ with the
help of the global time parameter. We prove that with any time-evolution of
measures, which respects the $N$-particle causal structure of
$\mathcal{M}_{\scriptscriptstyle (N)}$, one can associate a single measure on
the Polish space of '$N$-particle wordlines'. The latter is a
splitting-independent object, from which one can extract the evolution of
measures for any other global observer on $\mathcal{M}$. An additional asset of
the adopted measure-theoretic framework is the possibility to model the
dynamics of indistinguishable entities, such as quantum particles. As an
application we show that the multi-photon and multi-fermion Schr\"odinger
equations, although explicitly dependent on the choice of an external
time-parameter, are in fact fully compatible with the causal structure of the
Minkowski spacetime.
