Let $d\in\mathbb N$, $\tau>0$, $U\subseteq\mathbb R^d$ be open and $T_t$ be a $C^1$-diffeomorphism from $U$ onto an open subset of $\mathbb R^d$ for $t\in[0,\tau)$ with $T_0=\operatorname{id}_U$. Note that $$V:=\bigcup_{t\in[0,\:\tau)}T_t(U)$$ is open. Let $\lambda$ denote the Lebesgue measure on $\mathcal B(\mathbb R)$ and $$\mathcal F(\Omega):=\int_\Omega f\:{\rm d}\lambda^{\otimes d}\;\;\;\text{for }\Omega\in\mathcal B(U)$$ for some $f\in L^1\left(\left.\lambda^{\otimes d}\right|_V\right)$. Now let $$g(t,x):=\left|\det{\rm D}T_t(x)\right|(f\circ T_t)(x)\;\;\;\text{for }(t,x)\in[0,\tau)\times U,$$ $\Omega\in\mathcal B(U)$, $\Omega_t:=T_t(\Omega)\in\mathcal B(V)$ for $t\in[0,\tau)$ and $$G(t):=\int_\Omega g(t,x)\:\lambda^{\otimes d}({\rm d}x)\;\;\;\text{for }t\in[0,\tau).$$ By the substitution rule $$\mathcal F(\Omega_t)=G(t)\;\;\;\text{for all }t\in[0,\tau).\tag1$$
I'd like to find suitable assumptions ensuring that $[0,\tau)\ni t\mapsto\mathcal F(\Omega_t)$ is differentiable at $0$. Moreover, I would like to find further assumptions allowing me write the derivative as an integral over the boundary of $\Omega$.
Ignoring rigor for a moment: Assume there is a $v:[0,\tau)\times V\to\mathbb R^d$ with $$v(t,T_t(x))=\frac{\partial T}{\partial t}(t,x)\;\;\;\text{for all }(t,x)\in(0,\tau)\times U\tag2.$$ Then (still ignoring rigor) $$\left(\frac\partial{\partial t}{\rm D}T\right)(t,x)={\rm D}v(t,T_t(x)){\rm D}T_t(x)\tag3$$ and hence $$\frac\partial{\partial t}(\det{\rm D}T)(t,x)=(\nabla\cdot v)(t,T_t(x))\det{\rm D}T_t(x)\tag4$$ for all $(t,x)\in(0,\tau)\times U$. Now, if $[0,\tau)\times U\ni(t,x)\mapsto{\rm D}T_t(x)$ is continuous in the first argument, then (since $\det{\rm D}T_0=\det\operatorname{id}_{\mathbb R^d}=1>0$) there is a $\delta\in(0,\tau]$ such that $\det{\rm D}T_t>0$ for all $t\in[0,\delta)$.
How can we make these arguments rigorous and include?
Remark: I've found the following claim in this book, but I don't get how they get the differentiability of $t\mapsto J_t$ from their assumptions:
