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.
Assume $$(0,\tau)\times U\ni(t,x)\mapsto T_t(x)\tag1$$ is continuously differentiable in the first argument and let $v:(0,\tau)\times V\to\mathbb R^d$ be differentiable in the second argument with $$v(t,\;\cdot\;)\circ T_t=\frac{\partial T}{\partial t}(t,\;\cdot\;)\tag2$$ for all $t\in(0,\tau)$.
By the chain rule, it is easy to see that $(2)$ is differentiable in $x$ and $${\rm D}\left(v(t,\;\cdot\;)\circ T_t\right)(x)={\rm D}v\left(t,T_t(x)\right)\circ{\rm D}T_t(x)\tag3$$ for all $(t,x)\in(0,\tau)\times U$.
Can we show that $$(0,\tau)\times U\ni(t,x)\mapsto{\rm D}T_t(x)\tag4$$ is differentiable in the first argument and $$\left(\frac\partial{\partial t}{\rm D}T\right)(t,x)=\left({\rm D}\frac{\partial T}{\partial t}\right)={\rm D}\left(v(t,\;\cdot\;)\circ T_t\right)(x)\tag5$$ for all $(t,x)\in(0,\tau)\times U$?