I am looking for natural and interesting examples of stochastic processes with convergent finite-dimensional distributions but no tightness. As for the space, it is either $C[0,1]$ equipped with the uniform norm or $D[0,1]$ equipped with the Skorokhod topology $M_2$. Here I've chosen the topology $M_2$ since it is weaker than the Skorokhod topologies $J_1$ and $M_1$, so tightness in $M_2$ is the least restrictive.

What I am up to are examples of interest by their own and not specifically crafted to answer the question. Say, some type of stochastic processes that was the subject of a reasonable paper, where one of the main results proves only convergence of fi-di distributions, at least in some cases.

**UPD:** My best example is on partial sums of moving averages of heavy-tailed i.i.d. random variables, where the coefficients have different signs. The main point of this example is that it combines naturally few modes of tightness and non-tightness in various Skorokhod topologies.

Let $(X_n)_{n \ge 0}$ be i.i.d. r.v.'s from the domain of normal (for simplicity) attraction of an $\alpha$-stable law with $\alpha \in (0,2)$. Consider the sequence of moving averages $Y_n:=a_0 X_n + a_1 X_{n-1}$ and the processes of their partial sums $S_n(t):= \sum_{k=1}^{[nt]} Y_k$, where $t \in [0,1]$. Put $a:=a_0 + a_1$ and assume that $a \neq 0$.

Then $S_n(t) / n^{1/\alpha}$ converges weakly to $a S(t)$ for every $t$, where $S$ is the limiting $\alpha$-stable process. The same clearly holds for all fi-di distributions. Then:

-If $a_0 \neq 0, a_1= 0$, then $S_n / n^{1/\alpha}$ converges weakly to $aS$ in $(D, J_1)$.

-If $a_0 \neq 0, a_1 \neq 0$, then the distributions of $S_n / n^{1/\alpha}$ are not $J_1$-tight.

-However, if $a_0 a_1 \ge 0$, then $S_n / n^{1/\alpha}$ converge weakly to $aS$ in $(D, M_1)$ (Avram and Taqqu, 1992).

-If $a_0 a_1 <0 $, then the distributions of $S_n / n^{1/\alpha}$ are not even $M_2$-tight.

There are examples of moving averages of higher orders where there is $M_2$-convergence (with proofs due to Basrak and Krizmanic, 2014).