pr.probability – LLN for variable that is a function of n

Let $x sim P$ be a random variable and $f_k(x)$ be a random function of $x$ with parameter $k$. Further, let $k$ be a function of n such that $lim_{n to infty}k=infty$ and $lim_{n to infty} frac kn=0$. $f_k(x)$ is such that by dominated convergence, it holds that
begin{align*}
lim_{k to infty} E(f_k(x))=E(lim_{n to infty}f_k(x))=E(f(x))
end{align*}

Now I want to show that
begin{align*}
lim_{n to infty} frac 1n sum_{i=1}^nf_k(x_i)=E(f(x))
end{align*}

What I would like to do is something like this:
begin{align*}
lim_{k to infty}lim_{n to infty}frac 1n sum_{i=1}^nf_k(x_i)=lim_{k to infty}E(f_k(x))=E(f(x))
end{align*}

Additionally it holds that $f_k(x_i), f_k(x_j)$ might have a bit of a dependency structure. Sorry for not being clearer on this issue. However, $f(x_i), f(x_j)$ are independent.