Let $X=(X_t)_{t\in I}$ ($I\subset\mathbb{R}$ a (non-degenerate) interval) be a stochastic process with continuous sample paths and such that $X_t$ admits a continuous Lebesgue density $\chi_t\in C(\mathbb{R}^d)$ for each $t\in I$.
Notation: Given $\delta>0$ and $t_0\in I$, we write $I_\delta(t_0):= I\, \cap \,(t_0-\delta, t_0+\delta)$.
Are you aware of (minimal) sufficient conditions on $X$ which guarantee that for each $K\subset\mathbb{R}^d$ compact the following implication holds:
$$\tag{$\star$}\min_{x\in K}\chi_{t_0}(x) > 0 \quad(\text{some }t_0\in I)\ \quad \Longrightarrow \quad \exists\,\delta>0 \, : \, \inf_{t\in I_\delta(t_0)}\,\min_{x\in K}\chi_{t}(x) > 0.$$
(A sufficient condition for $(\star)$ would be that $(t,x)\mapsto\chi_t(x)$ is continuous, but it is not clear how this translates to a natural condition on $X$.)
