Integro-local theorems for sums of independent random vectors in the series scheme

Let $S(n)=\xi(1)+\dots+\xi(n)$ be a sum of independent random vectors $\xi(i)=\xi_{(n)}(i)$ with general distribution depending on a parameter $n$. We find sufficient conditions for the uniform version of the integro-local Stone theorem to hold for the asymptotics of the probability $\mathsf P(S(n)\in\Delta[x))$, where $\Delta[x)$ is a cube with edge $\Delta$ and vertex at a point $x$.