I'm trying to read chapter 8 of the book on gradient flows by Ambrosio-Gigli-Savaré. In this context, I would like to better understand how the theory works for the following specific example. Take the family of probability measures on the real line $$\mu_t=(1-t)\delta_0+t\delta_1, \quad t\in(0,1),$$ where $\delta_x$ denotes the Dirac delta at $x$. It seems that this probability-valued curve is absolutely continuous with respect to the Wasserstein metric, but it does not seem to satisfy the conclusions of Theorem 8.3.1 in the book. In other words, there does not seem to be a vector field satisfying the continuity equation for this family of probabilities. As far as I can see the proof would already fail at equation (8.3.10), since it is possible to produce a test function with $\partial_t\phi\neq 0$ everywhere, yet $\partial_x\phi=0$ on the support of $\mu_t$, namely, $\{0,1\}$.

So my question is, what am I missing here?

Thanks a lot in advance.