Let $\Gamma_t$ be a compact hypersurface for each $t \in [0,T]$. Let $Q=\Gamma_0\times(0,T)$ and $$Q_T :=\bigcup_{t \in (0,T)} \Gamma_t \times \{t\}.$$
For each $t \in [0,T]$, suppose that $F^t_0:\Gamma_t \to \Gamma_0$ is a $C^k$-diffeomorphism with inverse $F^0_t:\Gamma_0 \to \Gamma_t$.
We are given functions $\tilde u_n: Q \to \mathbb{R}$ and $\tilde u:Q \to \mathbb{R}$ such that $$\tilde u_n\to \tilde u\quad \text{pointwise a.e. in $Q$}.$$ I want to show that the functions $u_n$ satisfy $$\tilde u_n(F^{(\cdot)}_0, \cdot) \to \tilde u(F^{(\cdot)}_0, \cdot)\quad\text{pointwise a.e. in $Q_T$,}$$ i.e. that the pushforwards of sequence converges to the pushforward of the limit pointwise a.e.
Incomplete proof
Now we are given that for all $(x,t) \in Q\backslash Z$ (where $Z \subset Q$ is a set of measure zero) and every $\epsilon > 0$, there exists an $N$ such that if $n > N$, we have $$|\tilde u_n(x,t) - \tilde u(x,t)| < \epsilon$$ which can be written as $$|\tilde u_n(F^t_0(y),t) - \tilde u(F^t_0(y),t)| < \epsilon$$ where $y = F^0_t(x)$.
Now I don't know what to do.. how do I map null sets of $Q$ to null sets of $Q_T$ in the right way to achieve the result?