What ways are known to correspond, or transfer, a Borel probability measure $\mu$ over some Banach space $X$ to a Borel probability measure $P$ over $X^{*}$, the dual space?
Of course, if $X^{*} = X$, e.g. if $X$ is Hilbert, then the correspondence is trivial.
Admittedly, this question might seem a bit vague, and one could reasonably want me to specify what properties I'd like the hypothetical correspondence to satisfy, but for now I'm just wondering what's already out there along these lines.
If you are satisfied with the Hilbert space situation, the natural thing to do is to generalize it. First, suppose that the norm of $X$ is uniformly convex and uniformly smooth. Then for every $x\in X$ there exists a unique $x^*\in X^*$ such that $\|x^*\|=\|x\|$ and $x^*(x)=\|x\|^2$. (Exists by Hahn-Banach, is unique by strict convexity of $X^*$.) This defines a map $J:X\to X^*$, which turns out to be a homeomorphism. Furthermore, both $J$ and $J^{-1}$ are uniformly continuous on bounded sets. This is a standard fact which may be found in section 12.2 of Nonlinear Functional Analysis by Deimling (now published by Dover). Using $J$, you can push forward measures from $X$ to $X^*$ and back.
If you are pushing only in the direction from $X$ to $X^*$, then you don't really need $J^{-1}$. The uniform smoothness of $X$ suffices to make $J$ continuous.
Without any assumptions on the norm of $X$, you have to confront the fact that $J$ may be multi-valued and discontinuous. For example, on $\ell_1$ it sends $(x_n)\in \ell_1$ to $(\|x\|_{1}\operatorname{sign}x_n)\in\ell_\infty$ with obvious issues concerning $\operatorname{sign}$. If your measure $\mu$ gives positive mass to some set on which $J$ is multivalued, it will not be clear how to distribute this mass over $X^*$.
Also, you may be worried about the Borel measurability of $J$. The good news is that $J$ can be made continuous by relaxing the requirement $\langle J(x),x\rangle=\|x\|^2$ by an arbitrarily small amount. See Theorem 4.11 in Banach space complexes by Ambrozie and Vasilescu (unfortunately not published by Dover), or here. This may be useful both for proving measurability (as pointwise limit of continuous functions) and for getting around the multivaluedness (push forward by some continuous approximate duality maps, and then take a weak limit of measures).