Suppose I have a sequence of (Markov) stochastic processes $\left\{X^n_t:n\in\mathbb N, t\in[0,T]\right\}$ constructed on a common probability space.

None of my $X^n_t$ are continuous in $t$, but there is an almost surely continuous (ItÃ´) process $X_t$ such that for any fixed, monotonic sequence $t_n\to t$ I have $X^n_{t_n} \to X_t$ almost surely.

I'd like to strengthen this result as far as possible, Ideally I'd like to be able to say that with probability one for every $t\in[0,T]$ and every sequence $t_n \to t$ I have $X^n_{t_n}\to X_t$.

Obviously the assumptions here are not enough. For every example in which the strong convergence statement holds I could choose a sequence of iid uniform $S_n\in[0,T]$ and set $X^n_t = 0$ whenever $S_n-t\in\mathbb Q$ which would destroy the result without affecting the assumptions.

My question is, what else do I need to do to get the result?

Is there a known necessary condition or proof schema for aproaching this problem?