Suppose that $f_n: \mathbb{R} \times [0,\infty) \to \mathbb{R}$ is a uniformly bounded sequence (i.e., there exists $C>0$: $|f_n| < C$ for every $n$) such that $$ f_n \in C^2_x \times C^1_t, $$ and, as $n \to \infty$, \begin{equation} \tag{1} f_n(x,t) \longrightarrow f(x,t), \quad \text{a.e.} -(x,t) \in \mathbb{R} \times [0,\infty). \end{equation} Let $x_n: [0,\infty) \to \mathbb{R} $ be a sequence of absolutely continuous functions that satisfies \begin{equation} \tag{2} \frac{d}{dt} x_n(t) = f_n(x_n(t),t), \quad \text{a.e.} - t, \; x_n(0)=0, \end{equation} and \begin{equation} \tag{3} x_n(\cdot) \to x(\cdot), \quad \text{uniformly in any bounded interval in } [0,\infty), \end{equation} for some function $x: [0,\infty) \to \mathbb{R} $.

**Question.**

- Is the following statement true: As $n \to \infty$, \begin{equation} \tag{4} f_n(x_n(t),t) \longrightarrow f(x(t),t), \quad \text{a.e.} - t \in [0,\infty)? \end{equation}
- Is (4) true for any $x_n$ not necessarily satisfies (2)?
- If (4) is wrong, what are additional requirements needed on $f_n$ to make it true?

**Remark.**

- The limit function $f$ does not have to be $C^1$, even $C^0$.
- The conclusion (4) is true if (1) is true uniformly, without the assumption (2).

**Updated.** The second question is answered negatively thanks to the counterexample by Iosif Pinelis.