In this article, we extend the foundations of the theory of differential inclusions in the space of probability measures recently laid down 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 prove new variants of the Filippov theorem, compactness of solution set and relaxation theorem for continuity inclusions studied in the Cauchy-Lipschitz framework. We also propose an existence result ``à la Peano'' for this class of dynamics, under Carathéodory-type regularity assumptions. The latter is based on a set-valued generalisation of the semi-discrete Euler scheme originally proposed by Filippov to study ordinary differential equations with measurable right-hand sides.