Suppose $G$ is a locally compact group, $X$ is a locally compact space equipped with a nice right action $r_g$ of $G$ (I suspect every orbit being compact qualifies as "nice" here, but this probably isn't the greatest generality possible), and $(V,\rho)$ is a finite-dimensional left representation of $G$. Is there any characterization, in terms of some kind of data on the orbit space, of locally-defined $V$-valued measures on $X$ (in the sense of a system of measures on every compact subset of $X$ that behave well under restriction) which respect $G$'s action in the sense that $\left(\left(r_g\right)_*\mu\right)(E)=\rho_g\mu(E)$ for every precompact Borel set $E$? What if we let $V$ be a Banach space or similar? In the case where $X=G$ and $V$ is trivial, these are just Haar measures, but I've been unable to find any discussion of this sort of problem in greater generality.
(Here, all spaces are Hausdorff and all measures are Radon.)