= Solution
The shear $(t,x,v)\mapsto(t,x+tv,v)$ is a smooth <diffeomorphism> with <determinant> one and sends compact sets to compact sets. Thus $F,G$ are <locally integrable functions>. On a compact time interval $I$ and compact label set $K\subset\mathbb R^6$, the <integral> $\int_K\int_I|G|\,dt\,dx\,dv$ is finite. The <Fubini theorem>, followed by a countable exhaustion of time intervals and label sets, shows that $G(\cdot,x,v)\in L^1_{\rm loc}(\mathbb R)$ for almost every $(x,v)$.
To transform the weak equation, choose $\phi(t,x,v)=\rho(x-tv,v)\psi(t)$, where $\rho\in C_c^\infty(\mathbb R^6)$ and $\psi\in C_c^\infty(\mathbb R)$. Including a velocity cutoff in $\rho$ ensures the required <compact support>. The transport <derivative> of this <test function> is $\rho(x-tv,v)\psi'(t)$. Change variables to the characteristic labels to obtain
$$
\int\rho(x,v)\left[\int\bigl(F(t,x,v)\psi'(t)+G(t,x,v)\psi(t)\bigr)\,dt\right]dx\,dv=0.
$$
The fundamental test-function identity makes the bracket zero almost everywhere. Choose a countable dense family of time <test functions> on each bounded interval and extend by continuity; there is then a single negligible label set outside which $\partial_tF=G$ as one-dimensional distributions.
For each remaining label, subtract the primitive $H(t)=\int_0^tG(s)\,ds$. The <distributional derivative> of $F-H$ is zero, so it is a constant almost everywhere. Choose the resulting <absolutely continuous representative along free characteristics>, $\widetilde F(t)=C+H(t)$. The <fundamental theorem of calculus> gives
$$
\boxed{\widetilde F(t_2,x,v)-\widetilde F(t_1,x,v)=\int_{t_1}^{t_2}G(s,x,v)\,ds\quad\text{for all }t_1,t_2}.
$$
The representative qualification is essential: an arbitrary locally integrable representative can be modified on one time slice without changing the distributional equation, and would not satisfy the identity for every pair of times.
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