= Solution
For each $p\geq1$ with $f_0\in L^p$, a sufficient condition is $h\in L^1([0,T];L^p(\mathbb R^2))$ on every finite interval. The <Minkowski integral inequality> and measure preservation give the <finite-time Lp bound for Hamiltonian transport>
$$
\boxed{\|f_t\|_p\leq\|f_0\|_p+\int_0^t\|h(s)\|_p\,ds<\infty.}
$$
For all finite $p$ simultaneously, impose, for example, $h\in L^1_{\rm loc}([0,\infty);L^1\cap L^\infty)$. The elementary bound $\|h_s\|_p\leq\|h_s\|_1^{1/p}\|h_s\|_\infty^{1-1/p}$ and the <Holder inequality> in time show that its $L^p$ <norms> are locally integrable for every $1\leq p<\infty$. If a bounded initial value is also required, the same assumption controls the $p=\infty$ endpoint by the previous part.
A concrete stronger condition, compatible with a nonzero $C^1$ source, is that on each finite time interval the source has a common <compact support> in $(x,v)$. Its continuity then makes it bounded on that compact cylinder, so all these integrability conditions hold. A nonzero smooth source compactly supported in phase space supplies examples.
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