The answer is No, at least if your definition of a "nice" distribution includes anything in the convex hull of distributions that arise from time averages.

Time averages from a steady rotational flow on the circle $S^1$ will produce Lebesgue measure $\mu$. Time averages from a flow toward an attracting fixed point at the north pole $P$ will produce a point mass $\delta_P$.

But there is no flow on $S^1$ whose time averages lead to $\frac{1}{2}(\mu + \delta_P)$. (Consider the orbit of $P$.)

You can generalize this to flows on $\mathbb{R}^n$ by embedding $S^1$ as a global attractor, and generalize to continuous maps by replacing the rotational flow with an irrational rotation.

Let $\mu$ be Lebesgue measure on $S^1$, and $\delta_P$ be a point-mass at a point $P \in S^1$.

Then there is no flow on $S^1$ whose time averages lead to $\frac{1}{2}(\mu + \delta_P)$. (Consider the orbit of $P$.)

This distribution seems like it should count as "well-behaved." Its support is connected, and both $\mu$ and $\delta_P$ themselves arise as time-averages (by a rotational flow, and a flow to an attracting fixed point, respectively).

You can generalize to $\mathbb{R}^n$ by looking at an embedded $S^1$, and generalize to continuous maps by replacing the rotational flow with an irrational rotation.