Global sections of pull-back of a line bundle.

Question

Let $\pi \colon \mathcal{X}\to T$ be a smooth and proper morphism of complex manifolds (say $T$ connected). Let $\mathcal{L}$ be a holomorphic line bundle on $\mathcal{X}$, and consider the pushforward $\pi_{*}\mathcal{L}$. Fix a point $t_0$ in $T$ and consider tha stalk $(\pi_{*}\mathcal{L})_{t_0}$, which in this case should be isomorphic to $\mathcal{L}(\mathcal{X}_{t_0})=\lim_{\substack{\longrightarrow\\\mathcal{X}_{t_0}\subset U}}\mathcal{L}(U)$. Does in this case $\mathcal{L}(\mathcal{X}_{t_0}) \cong \iota^{*}\mathcal{L}(\mathcal{X}_{t_0})$, where $\iota \colon \mathcal{X}_{t_0} \hookrightarrow \mathcal{X}$?