Let $f_n:\mathbb{R}^2\to\mathbb{R}^+$ and $h:\mathbb{R}^2\to\mathbb{R}^+$, such that $f_n(x_1,x_2)\to f(x_1,x_2)$, in order to apply the dominated convergence theorem to obtain $$ \lim_{n\to\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f_n(x_1,x_2)h(y,z)dydzdx_1dx_2=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f(x_1,x_2)h(y,z)dydzdx_1dx_2, $$ what do I have to verify about the functions in the integrand?
Well, I know I have to verify that $f_n$ is bounded for all $n$ and $(x_1,x_2)$ and the RHS integration must be finite, but is that it? or do I have to verify something else?
DCT applies to integration w.r.t any measure. Your multiple integrals are just integrals w.r.t Lebesgue measure on $\mathbb R^{4}$. So you need the usual assumption that $|f_n(x_1,x_2)h(y,z)| \leq |\phi (x_1,x_2,y,z)|$ with $|\phi (x_1,x_2,y,z)|$ integrable over $\mathbb R^{4}$.
(Since Lebesgue measure is an infinite measure uniform boundeness of $f_n$'s is not good enough).