Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $(\mathcal F_t)_{t\ge0}$ bea filtration on $(\Omega,\mathcal A)$ and $(X_t)_{t\ge0}$ be a process on $(\Omega,\mathcal A,\operatorname P)$. Remember that $X$ is called $\mathcal F$-Lévy in law if $X$ is $\mathcal F$-adapted, continuous in probability, $X_0=0$ and \begin{align}X_{s+t}-X_s&\text{ is independent of }\mathcal F_s\tag1\\X_{s+t}-X_s&\text{ has the same distribution as }X_t\tag2\end{align} for all $s,t\ge0$. Now let $(X^n_t)_{t\ge0}$ be a process on $(\Omega,\mathcal A,\operatorname P)$ for $n\in\mathbb N$ with $$X^n_t\xrightarrow{n\to\infty}X_t\;\;\;\text{in probability for all }t\ge0\tag3.$$
If $X^n$ is $\mathcal F$-Lévy in law for all $n\in\mathbb N$, then $(3)$ implies that $X$ satisfies $(1)$ and $(2)$.
However, if each $X^n$ is Lévy in law with respect to its generated filtration $\left(\mathcal F^{X^n}_t\right)_{t\ge0}$, can we still show that $X$ satisfies $(1)$ with $\mathcal F$ replaced by $\mathcal F^X$? Moreover, will $X$ inherit the remaining properties to be $\mathcal F^X$-Lévy in law?