Let ${\mathcal{X}}$, ${\mathcal{X}}^{\prime}$ be proper dlt models of $X$, with ${\mathcal{X}}^{\prime}$ dominating ${\mathcal{X}}$ via a proper birational morphism $\rho\colon{\mathcal{X}}^{\prime}\to{\mathcal{X}}$. Let ${\mathcal{L}}^{\#}=({\mathcal{L}},\psi_{0})$ be a residually metrized model of $K_{X}$ consisting of a model ${\mathcal{L}}$ of $K_{X}$ determined on ${\mathcal{X}}$ and a continuous metric $\psi_{0}$ on ${\mathcal{L}}_{0}$. Set ${\mathcal{L}}^{\prime}=\rho^{*}{\mathcal{L}}$, $\psi^{\prime}_{0}=\rho^{*}\psi_{0}$ and ${\mathcal{L}}^{\prime\#}=({\mathcal{L}}^{\prime},\psi^{\prime}_{0})$. We must prove that $\mu_{{\mathcal{L}}^{\prime\#}}=\mu_{{\mathcal{L}}^{\#}}$.

Let $\sigma^{\prime}$ be a top-dimensional face of $\Delta({\mathcal{L}}^{\prime})$, $Y^{\prime}$ the associated stratum of ${\mathcal{X}}^{\prime}_{0}$, $Y$ the minimal stratum of ${\mathcal{X}}_{0}$ containing $\rho(Y^{\prime})$ and $\sigma=\sigma_{Y}$ the associated simplex of $\Delta({\mathcal{X}})$. Then $\sigma$ and $\sigma^{\prime}$ have the same dimension, and if we (somewhat abusively) identify $\sigma$ and $\sigma^{\prime}$ with their images in $\operatorname{Sk}(\phi_{\mathcal{L}})\subset X^{\mathrm{an}}$, then $\sigma^{\prime}$ is a rational subsimplex of $\sigma$. It suffices to prove that $\mu_{{\mathcal{L}}^{\prime\#}}(\sigma^{\prime})=\mu_{{\mathcal{L}}^{\#}}(\sigma^{\prime})$.

Now $\rho$ restricts to a birational morphism of $Y^{\prime}\to Y$, so since $\lambda_{\sigma}|_{\sigma^{\prime}}=\lambda_{\sigma^{\prime}}$ and $b_{\sigma}=b_{\sigma^{\prime}}$, it suffices to prove that $\operatorname{Res}_{Y^{\prime}}({\mathcal{L}}^{\prime\#})=\rho^{*}\operatorname{Res}_{Y}({\mathcal{L}}^{\#})$. But this is formal. Indeed, we have $(\rho|_{Y})_{*}(B_{Y^{\prime}}^{{\mathcal{L}}^{\prime}})=B_{Y}^{\mathcal{L}}$ and we can identify $(\rho|_{Y})^{*}K_{(Y,B_{Y}^{\mathcal{L}})}$ with $K_{(Y^{\prime},B_{Y^{\prime}}^{{\mathcal{L}}^{\prime}})}$ in such a way that the restriction of $\psi^{\prime}_{0}$ to ${\mathcal{L}}^{\prime}|_{Y^{\prime}}=K_{(Y^{\prime},B_{Y^{\prime}}^{{\mathcal{L}}^{\prime}})}$ coincides with the pullback under $\rho|_{Y^{\prime}}$ of the restriction of $\psi_{0}$ to ${\mathcal{L}}|_{Y}=K_{(Y,B_{Y}^{\mathcal{L}})}$. ∎