Consider the following situation:
$S, T$ are standard Borel spaces (say $S = [0,1]^k$, $T = [0,1]$ if it is helpful).
There is a a random variable $\zeta: \Omega \to S$.
$f_n(\zeta) \to^d \eta$, i.e. $f_n(\zeta)$ converges in distribution to some random variable $\eta$.
Will there always exist
A random variable $\zeta': \Omega' \to S$ (i.e. possibly with a different base space than $\Omega$), and
functions $g_n, g: S \to T$
$\zeta$ and $\zeta'$ have the same distribution,
for all $n$, $g_n(\zeta')$ has the same distribution as $f_n(\zeta)$, and
$g_n(\zeta') \to g(\zeta')\ a.e$, i.e. $g_n(\zeta')$ converges to $g(\zeta')$ almost everywhere (and in particular $g(\zeta')$ has the same distribution as $\eta$).