By [Gub98, Theorem 7.12] (see also [BFJ16, Corollary 2.3]), model metrics are dense in the set of continuous metrics on $K_{X}^{\mathrm{an}}$. Hence we may assume $\psi$ is a model metric. Using (5.3), it is enough to show that $\kappa^{\prime}(v^{\prime})=m\kappa(p(v^{\prime}))$ for a divisorial valuation $v^{\prime}\in X^{\prime\mathrm{div}}$. Let ${\mathcal{X}}$ be an snc model with $p(v^{\prime})\in\operatorname{Sk}({\mathcal{X}})$, and such that $\psi=\phi_{\mathcal{L}}$ for a model ${\mathcal{L}}$ of $K_{X}$ on ${\mathcal{X}}$. Since the normalized base change ${\mathcal{X}}^{\prime}$ of ${\mathcal{X}}$ is toroidal, we can choose a toroidal modification ${\mathcal{X}}^{\prime\prime}\to{\mathcal{X}}^{\prime}$ with ${\mathcal{X}}^{\prime\prime}$ snc. The induced morphism $\rho\colon{\mathcal{X}}^{\prime\prime}\to{\mathcal{X}}$ is toroidal; hence it satisfies the log ramification formula

$$mK^{\mathrm{log}}_{{\mathcal{X}}^{\prime\prime}/S^{\prime}}=\rho^{*}K^{\mathrm{log}}_{{\mathcal{X}}/S}.$$

By (5.7), we infer $\phi_{K^{\mathrm{log}}_{{\mathcal{X}}^{\prime\prime}/S^{\prime}}}-\psi^{\prime}=p^{*}(\phi_{K^{\mathrm{log}}_{{\mathcal{X}}/S}}-\psi)$, which gives the desired result since $v^{\prime}\in\operatorname{Sk}({\mathcal{X}}^{\prime\prime})$, $p(v^{\prime})\in\operatorname{Sk}({\mathcal{X}})$ imply $A_{{\mathcal{X}}^{\prime\prime}}(v^{\prime})=A_{{\mathcal{X}}}(p(v^{\prime}))=0$. ∎