Quantum Fisher Information of an infinitesimal evolution of the system Does it make some sense to consider the Quantum Fisher Information $F_Q$ of the dynamics of a density matrix $\dot{\rho}(t)$?
In other words,
$$
F_Q[\,\dot\rho(t)\,] = \mathrm{Tr}[\,\partial_\theta\dot\rho(t)\,L_\theta\,]
$$
where $\theta$ is the parameter that is estimated and $L_\theta$ the symmetric logarithmic derivative associated to $\dot\rho(t)$ which is solution of $\partial_\theta \dot\rho(t) = \frac{1}{2}\big(L_\theta\,\dot\rho(t) + \dot\rho(t)\,L_\theta\big)$.
If yes, what is the meaning of such a quantity? What is then the difference between the latter one and the quantum information flow $\frac{\mathrm{d}}{\mathrm{d}t}F_Q[\,\rho(t)\,]$?
 A: The quantum Fisher information is (almost everywhere) equivalent to the Bures metric, with
$$[d_{\mathrm{B}}(\rho,\rho+d\rho]^2\propto F_Q(\rho)d\theta^2$$  and its multiparameter generalization to quantum Fisher information matrices.
We can thus interpret the quantum Fisher information as telling us how much the density matrix changes to changes in the underlying parameter(s) $\theta$. In turn, replacing $\rho$ with $\dot{\rho}$ in the quantum Fisher information will yield a quantity governing the curvature of how $\dot{\rho}$ varies with changes in the underlying $\theta$.
We can gain more insight by taking the time derivative of the equation for the regular symmetric logarithmic derivatives, which I'll call $L$, and I'll refer to the $L$ defined in the question as $\tilde{L}$:
\begin{align}
\partial_\theta \dot{\rho}_\theta&\equiv\frac{\tilde{L}\dot{\rho}+\dot{\rho} \tilde{L}}{2}\\
\partial_\theta {\rho}_\theta&\equiv\frac{L\rho+\rho L}{2}\\
\Rightarrow
\partial_\theta \dot{\rho}_\theta&=\frac{{L}\dot{\rho}+\dot{\rho} {L}+\dot{L}{\rho}+{\rho} \dot{L}}{2}.
\end{align}
The QFI for the time derivative is equal to
$$F_Q( \dot{\rho}_\theta)=\mathrm{Tr}\left(\partial_\theta\dot{\rho}_\theta \tilde{L}\right)
=\frac{1}{2}\mathrm{Tr}\left(\dot{\rho}_\theta\left\{L,\tilde{L}\right\}+ {\rho}_\theta\left\{\dot{L},\tilde{L}\right\}\right)$$
and the time derivative of the QFI is equal to
$$\partial_t F_Q({\rho}_\theta)=\mathrm{Tr}\left(\dot{\rho}_\theta {L}^2+{\rho}_\theta \left\{\dot{L},{L}\right\}\right),$$ where I have used the anticommutator $\left\{\cdot,\cdot\right\}$. Can we relate these two?
When we subtract the two quantities, we find the difference
\begin{aligned}
2F_Q( \dot{\rho}_\theta)-\partial_t F_Q({\rho}_\theta)
=\mathrm{Tr}\left(\dot{\rho}_\theta\left\{L,\tilde{L}-L\right\}+ {\rho}_\theta\left\{\dot{L},\tilde{L}-L\right\}\right).
\end{aligned} Mathematically, this deviates from zero because the SLDs for the density matrix and its time derivative are different ($\tilde{L}\neq L$), but I do not have a good physical interpretation of this difference.
One physical case is easy to interpet. When the regular SLD is independent of time ($\dot{L}=0$), such that the way that the state changes with $\theta$ is independent of how it changes with time, we have the following, more explict equations:
\begin{aligned}
\partial_\theta \rho_\theta(t)=\frac{L_\theta\rho_\theta(t)+\rho_\theta(t)L_\theta}{2} \quad\Rightarrow\quad \tilde{L}_\theta=L_\theta.
\end{aligned}
Then (and only then?) we find that the QFI for the time derivative and the time derivative of the QFI coincide (up to a factor of 2). Perhaps we can thus interpret the difference between those two quantities as encompassing how much $L_\theta$ depends on time; i.e., the difference tells us how much [the curvature of how $\rho_\theta(t)$ varies with $\theta$] depends on time.
