Let $(\mathcal{X},\mathcal{L})$ be a model of $(X_{\Sigma},L^{\otimes e})$ that realizes the algebraic metric $\|\cdot\|$. For short, denote $\mu=c_{1}(L,\|\cdot\|)\land\delta_{X_{\Sigma}}$ and $\mu_{\mathbb{S}}=c_{1}(L,\|\cdot\|_{\mathbb{S}})\land\delta_{X_{\Sigma}}$. By Proposition 5.55 there is a non-Archimedean field $H$ over $K$ and a semi-stable model ${\mathcal{X}}^{\prime}$ of $X_{\Sigma,H}$. We may further assume that all the components of the special fibre of ${\mathcal{X}}^{\prime}$ are defined over $H^{\circ}/H^{\circ\circ}$. Let $(L^{\prime},\|\cdot\|^{\prime})$ be the metrized line bundle obtained by base change to $H$. Then $(\|\cdot\|^{\prime})_{\mathbb{S}}$ is obtained from $\|\cdot\|_{\mathbb{S}}$ by base change. We denote by $\pi\colon X^{{\text{\rm an}}}_{\Sigma,H}\to X^{{\text{\rm an}}}_{\Sigma,K}$ the map of analytic spaces. Be will denote by $\mu^{\prime}$, $\mu^{\prime}_{\mathbb{S}}$, $\theta_{\Sigma}^{\prime}$ and $\rho_{\Sigma}^{\prime}$ the corresponding objects for $X_{\Sigma,H}$. Then, by Proposition 2.35 and Proposition 5.53,

$$\mu_{\mathbb{S}}=\pi_{\ast}\mu^{\prime}_{\mathbb{S}}=\pi_{\ast}(\theta^{\prime}_{\Sigma})_{\ast}(\rho^{\prime}_{\Sigma})_{\ast}\mu^{\prime}=(\theta_{\Sigma})_{\ast}(\rho_{\Sigma})_{\ast}\pi_{\ast}\mu^{\prime}=(\theta_{\Sigma})_{\ast}(\rho_{\Sigma})_{\ast}\mu.$$

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