I'm looking for some sources on the derivation of the continuity equation, I'd like to show that if I have certain initial mass distribution (let's say probabilistic) $\rho$ and if particles move according to some (time-dependent) vector field $v_t(x)$, then my mass distribution $\rho_t$ at time $t$ should satisfy $$ \partial_t \rho_t + \nabla_x (\rho_t v_t) = 0 \ldotp$$ I'm asking here because all the references I've found online were either vague, without full proofs or physics-oriented, so making physically relevant assumptions and using physics' language, while I'd like to look at this problem from a strictly mathematical point of view.
The most important part for me are assumptions necessary for uniqueness, because I'd like to use the equation as an equivalent of saying that the measure at time $t$ is a push-forward of the initial measure by an appropriate flow. I'll be most grateful for any references - it seems to me that this is bound to be thouroughly addressed in some book, I just can't seem to find it.
Okay, I've found a good reference (at least it has all the elements I was looking for), so I'm gonna leave it here in case anyone else is looking for something similar - I'm using a book by Ambrosio, Gigli and Savare "Gradient flows in metric spaces and in the space of probability measures" published by Birkhauser, chapter 8 in particular.