Let $f: X\to T$ be a flat family, and $\mathcal{F}_t$ is a vector bundle on $X_t$ for some $t\in T$. Can this $\mathcal{F}_t$ be extended to a vector bundle $\mathcal{F}$ on $f^{-1}(U)$ for some open neighborhood of $t$?

If moreover $\mathcal{E}$ is a vector bundle on $X$, and $\mathcal{F}_t$ is a subbundle of $\mathcal{E}_t$ on $X_t$, then can this $\mathcal{F}_t$ be extended to a subbundle $\mathcal{F}$ of $\mathcal{E}$ on $f^{-1}(U)$ for some open neighborhood of $t$?