Let $Y$ be a $d$-dimensional subvariety of $X$ and ${\overline{L}}_{i}=(L_{i},\|\cdot\|_{i})$, $i=0,\dots,d-1$, a collection of approachable metrized line bundles on $X$. For each $i$, let $(\|\cdot\|_{i,l})_{l\geq 0}$ be a sequence of semipositive smooth (in the Archimedean case) or algebraic (in the non-Archimedean case) metrics on $L_{i}^{{\text{\rm an}}}$ that converge to $\|\cdot\|_{i}$. Then the measures $\operatorname{c}_{1}(L_{0},\|\cdot\|_{0,l})\land\dots\land\operatorname{c}_{1}(L_{d-1},\|\cdot\|_{d-1,l})\wedge\delta_{Y}$ converge weakly to a measure on $X^{\text{\rm an}}$.