I write $S(\mathbb R^d)$ the space of Schwartz functions and $S'(\mathbb R^d)$ the space of tempered distributions.
Assume that I have a family $(u(t))_{t=0}^T \subset S'(\mathbb R^d)$ of tempered distributions such that for all fixed $\phi \in S(\mathbb R^d)$ : $$\langle u(t), \phi \rangle \in C^0([0,T])$$
Is it possible to find a sequence of Schwartz functions $(u_n(t,x))_{n \in \mathbb N} \subset S(\mathbb R^{1+d})$ such that for all fixed $t \in [0,T]$, $u_n(t, \cdot)$ converges weakly-$*$ to $u(t)$ ?
Here is an inconclusive idea :
One can prove using Banach-Steinhaus that for all $\phi \in S(\mathbb R^{1+d})$ : $$\langle u(t), \phi(t, \cdot) \rangle \in C^0([0,T])$$
So the family $(u(t))_{t=0}^T$ defines a $(1+d)$-dimensional tempered distribution $u$ using the following formula :
$$\langle u, \phi \rangle = \int_0^T \langle u(t), \phi(t, \cdot) \rangle dt$$
and we can find a sequence of Schwartz functions $(u_n(t,x))_{n \in \mathbb N} \subset S(\mathbb R^{1+d})$ such that
$$\langle u, \phi \rangle = \int_0^T \langle u(t), \phi(t, \cdot) \rangle dt = \lim \limits_{n \to +\infty} \int_0^T \langle u_n(t,\cdot), \phi(t, \cdot) \rangle dt = \lim \limits_{n \to +\infty} \langle u_n, \phi \rangle$$
From here, I couldn't go further. I have no guarantee that $u_n(t, \cdot)$ converges weakly-$*$ to $u(t)$.