Consider a sequence of $n-$dim Alexandrov spaces with curvature $\geq$ -1 $\{(M_i,p_i)\}$ Gromov-Hausdroff converging to an $n-$dim Alexandrov space $(M,p)$. Let $f:M\mapsto \mathbb R$ be a Lipschitz function, and $R>0$ fixed. For $0<r<R$, can we construct Lipschitz functions $f_i:B(p_i,R)\subset M_i\mapsto \mathbb R$ ,such that $\int_{B(p_i,r)}|\nabla f_i|^2d\mathcal H^n\to \int_{B(p,r)}|\nabla f|^2d\mathcal H^n$ ?

First note that if $B(p,R)$ lies in a distance chart then you can lift the function using the chart and it will satisfy the needed convergence.

Now assume you have few overlapping distance charts; they cover subset $\Omega$ and it might have nonempty complement $C=M\backslash \Omega$. You can lift the function in each chart and use partition of unity to glue these liftings together. Further you can extend the obtained function keeping it Lipschitz to whole $M_n$.

This produce a sequence of Lipschitz functions for which you have the needed convergence with an error depending on your Lipshcitz constant and volume of $C\cap B(p,R)$. The latter can be made as small as you want. So applying diagonal procedure you get the result.