On p.221 of Voisin's book on Hodge theory, there are two claims:
a) Let $B$ be a contractible smooth manifold. There exists a vector field $\chi$ on $B$ whose flow $\Phi_t$ is global and, given any neighborhood $U$ of the point we contract to, we can find $T$ such that, for all $t \geq T$, $Im_t$ is contained in $U$.
b) Let $B$ and $X$ be smooth manifolds, $\chi$ a global vector field on $B$, $\phi: X \rightarrow B$ a submersion [WAS MISSING, NOW ADDED] which is smooth and proper. Then $\chi$ lifts to a global vector field $\chi'$ on $X$. [as now stated see Need help understanding a lift of a vector field ] How do we prove these?