Derivatives in the space of finite signed measures. I just read about the Banach-space structure for the space of finite signed measures $\mathbf M(\mathcal B(\mathbb R^n)) =: \mathbf M$ with the norm defined via the total variation. 
Setting:
I want to write down a ODE for the family of measures on $\mathcal B(\mathbb R^n)$, which are defined by
$$
\rho_t := (\phi_t)_* \rho_0  \quad 
\Leftrightarrow \quad
\rho_t(A) := \rho_0( \varphi_t^{-1}(A)) \quad \forall A \in \mathcal B(\mathbb R^n) 
$$
where the map $\varphi_t : \mathbb R^n \to \mathbb R^n$ 
is a solution of an ODE
$$
\dot \varphi_t(x) = \mathbf v_1(x,t)
$$
for some at sufficiently smooth vector field $\mathbf v_1$.

Question:
Is it possible to write down an ODE for $\rho_t$?

My try to find a generator:
Of course, I could just try to calculate
$$
\frac{\mathrm d \rho_t}{\mathrm d t}(A) 
= \lim_{h \to 0} \frac{1}{h}
\left(
\rho_0( \varphi_{h}^{-1}(A) ) - \rho_0(A)
\right)
=
\lim_{h \to 0} 
 \frac{1}{h}
\int 1_{A}(\varphi_h(x)) - 1_{A}(x) \rho_0(\mathrm d x)
$$
but I don't know how to approach this term, since the mapping $\rho_0 : \mathcal B(\mathbb R) \to [-\infty,\infty] $ isn't too smooth.
Formally the limit might be
$$
\int_{\partial A} \mathrm{sign}( \langle \mathbf n , \mathbf v_1 \rangle_{\mathbb R^n} ) \, \rho_0(\mathbb d \mathbf n)
$$
but I am not sure how to get there and if this is correct and well-defined.
Any suggestions for related literature are also very appreciated!
Addition details on $\rho_0$:
In my setting, the measures are also assumed to have compact support,
since they model a bunch of particles and they are positive. I only consider to go into the space of signed measure due to the Banach-space structure.
 A: Just for reference:
An answer, which worked for me in the end, was to use a set of test function
$g \in C_c(\mathbb R^n)$ and use the pairing $\langle \rho_t, g \rangle = \int g \, \rho_t$.
For an empirical measure $\mu^N = \frac{1}{N} \sum_{i=1}^N \delta_{x_i}$ we can compute a weak form of the equations 
$$
\frac{\mathrm d}{\mathrm d t} \langle \rho_t , g \rangle 
= \frac{\mathrm d}{\mathrm d t} \frac{1}{N} \sum_i g(x_i(t)) 
= \frac{1}{N} \mathrm{D}_x g(x_i(t)) \dot x_i(t)
= \frac{1}{N} \mathrm{D}_x g(x_i(t)) v_1(x_i,t) \\
= \dots
= - \langle \mathrm{D}_x( v_1 \rho_t ) , g \rangle.
$$
The last equality has to be seen in a distributional sense. 
Then together with the Glivenko-Cantelli Lemma, which implies that empirical measures are dense in the space of measures with finite first moment, we can deduce that the equation has to hold weakly on the hole space of measures.
I found it for example in the article
On the Dynamics of Large ParticleSystems in the Mean Field Limit, François Golse
