Define $S = \{1,\dots, |S|\}$, and for each $s \in S$ let $\{X_{i,s}\}_{i=1}^\infty \overset{iid}{\sim} P_s$ and assume that $E[X_{i,s}] = 0$ for each $s$.
Now, for each $n$, define $n_s(n) \leq n$ with $n_s(n) \in \mathbb{N}$ and $\frac{n_s(n)}{n} \xrightarrow{P} \delta_s \in (0,1)$ with $\sum_{s\in S}n_s(n) = n$ and $\sum_{s\in S}\delta_s = 1$. Let $n_s(n)$ be non-decreasing for each $n$.
Now, I want to know whether \begin{align*} \frac{1}{\sqrt{n}}\begin{pmatrix} \sum_{i=1}^{n_1(n)}X_{i,1}\\ \vdots\\ \sum_{i=1}^{n_{|S|}(n)}X_{i,|S|} \end{pmatrix} \xrightarrow{d} N(0, \Sigma)~, \end{align*} for some covariance matrix $\Sigma$. Assume that $\{X_{i,s}\}_{i=1}^\infty$ independent of $\{X_{i,s'}\}_{i=1}^\infty$ for any $s'\neq s$.
I think I can show that each of the coordinate sums is asymptotically normal, as follows:
Fix some $s$ and define $X_i = X_{i,s}$. Now, let $Z_n = \frac{1}{\sqrt{n}}\sum_{i=1}^{n}X_{i}$, and now the CLT gives that, for some $\sigma_s$, $$Z_n \xrightarrow{d} N(0, \sigma_s)~,$$ and so $$E[I\{Z_n \leq x\}]=P(Z_n \leq x) \rightarrow \Phi_s(x)~,$$ where $\Phi_s(x)$ is the cdf of $N(0, \sigma_s)$.
Now, since $\frac{n_s(n)}{n} \xrightarrow{P} \delta_s$, we get the necessary result (by continuous mapping thm) if we can show that $E[E[I\{Z_{n_s(n)}\leq x\}| n_s(n)]] = E[I\{Z_{n_s(n)}\leq x\}] \rightarrow \Phi(x)$.
So, by the Vitali convergence theorem, we are done if we can show that $E[I\{Z_{n_s(n)}\leq x\}| n_s(n)] \xrightarrow{P} \Phi(x)$.
Finally,let $\epsilon > 0$ be given and note that there exists $N$ such that $|P(Z_n \leq x) - \Phi(x)| < \frac{\epsilon}{2}$ for all $n > N$. Now, we have that we can pick $n$ sufficiently large so that $P(n_s(n) \leq N) < \frac{\epsilon}{2}$, and so, for sufficiently large $n$, we have \begin{align*} P(|E[I\{Z_{n_s(n)}\leq x\}| n_s(n)] - \Phi_s(x)| > \epsilon) =&\ P(|E[I\{Z_{n_s(n)}\leq x\}| n_s(n)] - \Phi_s(x)| > \epsilon| n_s(n) > N)P(n_s(n) > N) + P(|E[I\{Z_{n_s(n)}\leq x\}| n_s(n)] - \Phi_s(x)| > \epsilon| n_s(n) \leq N)P(n_s(n) \leq N)\\ \leq&\ P(|E[I\{Z_{n_s(n)}\leq x\}| n_s(n)] - \Phi_s(x)| > \epsilon| n_s(n) > N)+P(n_s(n) \leq N)\\ <&\ \frac{\epsilon}{2} + \frac{\epsilon}{2}~, \end{align*} and so, $E[I\{Z_{n_s(n)}\leq x\}| n_s(n)] \xrightarrow{P} \Phi(x)$, as required.
I think this correct, but I am not sure how to to go from this to joint convergence. I think it should be possible to use the above result to say something like \begin{align*} \frac{1}{\sqrt{n}}\begin{pmatrix} \sum_{i=1}^{n_1(n)}X_{i,1}\\ \vdots\\ \sum_{i=1}^{n_{|S|}(n)}X_{i,|S|} \end{pmatrix}|\{n_1(n), \dots, n_s(n)\} \xrightarrow{d} N(0, \Sigma_2)~, \end{align*} where $\Sigma_2$ is the limit of some conditional covariance matrix? Or something like that? But I am not sure how to get there formally