Stability of pointed analytic worldsheets under restriction and base change

Restricting $\pi$ to ${\pi }^{-1}(B_0)$ preserves holomorphicity, surjectivity of its differential, the marked sections, and all local coordinate germs. For compact $K\subseteq B_0$, $K$ is compact in $B$, so ${\pi }^{-1}(K)$ is compact by properness of $\pi$; hence the restriction is proper. Its fibers and connected components are unchanged at points of $B_0$.

Fix $c\in \mathcal{C}$ with $\pi (c)=b$. In local holomorphic charts, write the base coordinates as $t_1,\dots ,t_m$ and the total-space coordinates as $u_1,\dots ,u_{m+1}$, since $\pi$ is a submersion with one-dimensional fibers. Surjectivity of $d{\pi }_c$ says at least one of the finitely many $m\times m$ minors of the complex Jacobian of $(t_1\circ \pi ,\dots ,t_m\circ \pi )$ is nonzero. Choose the lexicographically first such minor and reorder the total-space coordinates so its columns are $u_1,\dots ,u_m$, with the omitted coordinate named $u_{m+1}$. The holomorphic map $\mathrm{\Phi }(u)=(t_1\circ \pi (u),\dots ,t_m\circ \pi (u),u_{m+1})$ then has invertible complex differential. Viewed as a real map, its differential is invertible, so [F3] gives a local $C^1$ inverse. The inverse differential is the inverse of a complex-linear map and is therefore complex-linear at every point. Applying [F4] componentwise shows that this inverse is holomorphic. Thus these coordinates identify $\pi$ locally with projection $B\times D\to B$ for a disk $D\subset \mathbb{C}$. In these local product charts, ${\mathcal{C}}_f$ has coordinates $(b^{\mathrm{\prime }},u_{m+1})$ and projection ${\pi }_f$ is the projection to $b^{\mathrm{\prime }}$. The transition maps are holomorphic restrictions of the original maps and of $f$, so ${\mathcal{C}}_f$ is a complex manifold and ${\pi }_f$ is a holomorphic submersion.

Let $K\subseteq B^{\mathrm{\prime }}$ be compact. The fiber product is closed in $\mathcal{C}\times B^{\mathrm{\prime }}$ because $B$ is Hausdorff and the diagonal in $B\times B$ is closed. The inverse image ${\pi }_f^{-1}(K)$ is therefore closed in the compact set ${\pi }^{-1}(f(K))\times K$, since $f(K)$ is compact and $\pi$ is proper. Hence ${\pi }_f^{-1}(K)$ is compact, so ${\pi }_f$ is proper. The pulled-back sections are holomorphic and disjoint because the original sections are, and the coordinate maps $(z_i\circ {\mathrm{pr}}_{\mathcal{C}},{\pi }_f)$ are the pulled-back product charts. Each fiber is canonically the corresponding original fiber, so compactness and the marked-component condition are preserved.