Let $\mathcal{P}_2(\mathbb{R})$ be the space of probability measures with finite second moment on $\mathbb{R}$, and $(\mu_t)_{t\geq 0}\subseteq \mathcal{P}_2(\mathbb{R})^{\mathbb{R}^{\geq0 }}$ be a sequence of measures. I am interested in defining a time derivative $\partial_t \mu_t$.
This tutorial indicates that if there exists a sequence $(v_t)_{t\geq 0}\subseteq L_2(\mu_t)_{t\geq 0}$ such that for each $t$ and $\varphi \in C_c^\infty(\mathbb{R})$ we have $$\frac{d}{dt} \int \varphi d\mu_t= \langle \nabla\varphi, v_t\rangle_{\mu_t},$$ then $v_t$ is a velocity field of $\mu_t$, and in some sense is the definition of $\partial_t \mu_t$.
In my opinion, the most natural definition of $\partial_t\mu_t$ would be defined by $$\partial_t \mu_t=\lim_{\delta\to 0} \frac{\mu_{t+\delta}-\mu_t}{h},$$ so that $\partial_t \mu_t$ is a signed measure with zero total mass. Is this in some sense equivalent to the velocity field $(v_t)_{t\geq 0}$ above? Or is this there a reason why this quantity does not seem to be used in literature?