The notion of the weak derivative of a process of probability measures $(\mu_t)_{t\geq 0}$ in optimal transport theory seems to be so basic that, e.g., in "Optimal Transport, old and new" the author hardly bothers to even define it. Apparently it's somehow defined by "duality" with smooth functions, maybe somewhat like $\partial_t \int \phi \,d\mu = \int \phi'\,d(\partial_t \mu)$ for test functions $\phi$.
For some more context, it appears for example in the conservation of mass formula (which I cannot explain any further, unfortunately):
$$\partial_t \mu + \nabla \cdot (\mu \xi) = 0.$$
Where can I read about the definition of $\partial_t \mu$, what conditions are necessary and perhaps some explicit examples?
(As an aside I also wonder what $\mu \xi$ could mean for some "velocity field" (vector field?) $\xi$).