Given a topological group $G$, a $G$-space is a topological space $X$ equipped with an action of $G$, such that the map $(g,x) \mapsto g.x$ is continuous. The action is distal if no non-diagonal orbit of $G$ on $X \times X$ has an accumulation point on the diagonal, and minimal if every orbit is dense.

A theorem of Furstenberg (The Structure of Distal Flows, 1963) says that if $G$ is a locally compact group and $X$ is a compact metrizable $G$-space such that the action of $G$ is minimal and distal, then $X$ is built out of isometric extensions: that is, there is a (potentially transfinite) sequence $(X_{\beta})_{\beta \le \alpha}$ of compact $G$-spaces and surjective morphisms $X_{\beta+1} \rightarrow X_{\beta}$ so that $X_0$ is the one-point space, $X_{\alpha} = X$, $X_{\beta}$ is the inverse limit of $(X_{\gamma})_{\gamma < \beta}$ when $\beta$ is a limit ordinal, and for each successor ordinal $\beta+1$ there is a $G$-invariant real-valued function on $X_{\beta+1} \times X_{\beta+1}$ that restricts to a compatible metric on each fibre of $X_{\beta+1} \rightarrow X_{\beta}$. (Conversely, any $G$-space built out isometric extensions in this way is distal; this direction is much easier and doesn't need any special hypotheses.)

Subsequent authors have generalized this result by removing various hypotheses and obtaining a similar conclusion about the decomposition of distal minimal $G$-spaces; sometimes the $G$-invariant metrics need to be replaced with some version of equicontinuity, but this is just an inevitable consequence of allowing $G$-spaces that are not first-countable. One constant though seems to be that $X$ needs to be compact. Are there any similar results if $X$ is instead some 'nice' topological space, e.g. a Polish space or a locally compact Hausdorff space, but not necessarily compact? Alternatively, are there some instructive examples of distal minimal $G$-spaces that don't break into equicontinuous pieces?

I am mainly interested in the case when $X = G/K$ for some closed subgroup $K$ of a locally compact or Polish group $G$, if that makes life easier (so in particular, the action is transitive).