A Skorohod representation theorem for uniform distance

Let $\mu_n$ be a probability measure on the Borel $\sigma$-field on
$D[0,1]$ with respect to Skorohod distance, $n\geq 0$. Necessary and
sufficient conditions for the following statement are provided. On
some probability space, there are $D[0,1]$-valued random variables
$X_n$ such that $X_n\sim\mu_n$ for all $n\geq 0$ and
$\norm{X_n-X_0}\rightarrow 0$ in probability, where $\norm{\cdot}$
is the sup-norm. Such conditions do not require $\mu_0$ separable
under $\norm{\cdot}$. Applications to exchangeable empirical
processes and to pure jump processes are given as well.