  In this article, we derive conditions for the existence of solutions to
state-constrained continuity inclusions in Wasserstein spaces whose right-hand
sides may be discontinuous in time. These latter are based on a fine
investigation of the infinitesimal behaviour of the underlying reachable sets,
through which we show that up to a negligible set of times, every admissible
velocity of a continuity inclusion can be approximately realised as the metric
derivative of a solution of the dynamics, and vice versa. Building on these
results, we are able to establish necessary and sufficient geometric conditions
for the viability and invariance of stationary and time-dependent constraint
sets which involve a suitable notion of contingent cones in Wasserstein spaces,
and presented in ascending order of generality. We then close the article by
exhibiting two prototypical examples of constraints sets appearing in
applications for which one can compute relevant subfamilies of contingent
directions.
