  In this article, we extend the foundations of the theory of differential
inclusions in the space of probability measures with compact support, laid down
recently in one of our previous work, to the setting of general Wasserstein
spaces. Anchoring our analysis on novel estimates for solutions of continuity
equations, we propose a new existence result ``\`a la Peano'' for this class of
dynamics, under mere Carath\'eodory regularity assumptions. The latter is based
on a set-valued generalisation of the semi-discrete Euler scheme proposed by
Filippov to study ordinary differential equations with measurable right-hand
sides. We also bring substantial improvements to the earlier versions of the
Filippov theorem, compactness and relaxation properties of the solution sets of
continuity inclusions which are derived in the more restrictive
Cauchy-Lipschitz setting.
