<p>This paper focuses on differential inclusions for measures, that are differential relations whose solutions are time-evolving measures. The definition of evolution equations for measures attracted a lot of attention recently. We start by recalling the main concepts developed in the latest literature and comparing them. In particular, we show how the definition of Measure Differential Inclusion is the most general allowing to model phenomena as diffusion from a Dirac delta. Then we pass to Lyapunov-type stability proposing two concepts of stability, based on the measure support and first moment, and show relationships between such definitions depending on the assumptions on the evolution equation used.</p>