Let $X_1,X_2,\ldots,X_n$ be i.i.d. random variables in $\mathbb{R}$ with common cumulative distribution function (CDF) $F(x)$. The empirical approximation to $F(x)$ is defined as follows:

$$\hat{F}_n(x)=\frac{1}{n}\sum_{i=1}^n\mathbf{1}_{\{X_i\leq x\}}(x)\tag{1}$$

where $\mathbf{1}_A(x)$ is an indicator function: unity when $x\in A$ and zero otherwise.

The Glivenko-Cantelli Theorem states that $\hat{F}_n(x)$ converges to $F(x)$ uniformly, that is, $\sup_{x\in\mathbb{R}}|\hat{F}_n(x)-F(x)|\rightarrow 0$ almost surely as $n\rightarrow\infty$. The Dvoretzky–Kiefer–Wolfowitz inequality quantifies the rate of convergence as follows: $P(\sup_{x\in\mathbb{R}}|\hat{F}_n(x)-F(x)|>\epsilon)\leq2e^{-2n\epsilon^2}$.

I am wondering if anything is known about the distributions of the errors in the empirical approximation of a CDF: that is, how is the random variable $\varepsilon_n(x)\equiv\hat{F}_n(x)-F(x)$ distributed? Is there perhaps asymptotic Gaussianity here, maybe for some classes of $F(x)$?

Here is the reason for my question: suppose I draw $nM$ random variables i.i.d. that have the common CDF $F(x)$ where $m=1,2,\ldots,M$, and compute $M$ empirical approximations as follows:

$$\hat{F}_{n,m}(x)=\frac{1}{n}\sum_{i=n(m-1)+1}^{nm}\mathbf{1}_{\{X_i\leq x\}}(x)\tag{2}.$$

Now I need to approximate $F(x_0)$ for a given $x_0$. Which approach would yield a better estimate: the average $\frac{1}{M}\sum_{m=1}^M\hat{F}_{n,m}(x_0)$ over the approximations in (2), or using $nM$ samples in constructing the approximation in (1)? If it helps, the distribution that I am trying to approximate is a mixture of binomial random variables.