Consider an independent collection of random variables $W_i, i=1,\dots,n.$ and let $Z \sim N(0,1)$. Roughly speaking, we know that $W_i$ are close in distribution to $Z$, say each is itself a sum of $d_i$ independent (bounded) variables hence satisfies a Berry-Esseen type CLT of the form $$ d_W(W_i,Z) \lesssim \frac{1}{\sqrt{d_i}} $$ where $d_W$ is the $L_1$ Wasserstein distance. One would then hope that $\frac1{\sqrt n} \sum_{i=1}^n W_i$ is itself close to a standard normal in distribution. Are there quantitative bounds on how close this approximation is? Namely, can we get a bound on $$ \delta_n := d_W\Big(\frac1{\sqrt n} \sum_{i=1}^n W_i, Z\Big) $$ in terms of $d_W( W_i,Z)$? What sort of growth is allowed/required for $\max_i d_i$ and $\min_i d_i$ relative to $n$ to ensure that $\delta_n = o(1)$ as $n \to \infty$.

EDIT: We can assume that $W_i$ are zero-mean. A more general question is this: Is there a distance $d$ that metrizes the weak convergence (of probability measures) for which we have something close to $$ d \Big(\frac1{\sqrt n} \sum_{i=1}^n W_i, Z\Big) \le C \frac{\max_i d(W_i,Z)}{\sqrt{n}}, $$ for independent zero mean $W_i$. To see that this is plausible, consider the case where $W_i$ have equal variance, say $\mathbb E W_i^2 = 1$. Then a bound of the form $\lesssim \frac1{\sqrt{n}}\max_i \mathbb E |W_i|^3$ holds for Kolomogrov distance, maybe even for $d_W$. The question is whether equal variance $\mathbb E W_i^2 =1$ assumption can be relaxed and 3rd moment bounds can be replaced with the distance itself (the latter might be doable if $W_i$ are sufficiently concentrated, say they are sub-Gaussians). It might not be possible if Berry and Esseen bounds for non-identically distributed variables are tight.

A bonus point is if everything can be done in higher dimensions, say $W_i \in \mathbb R^m$ and $Z \sim N(0,I_m)$.