  We consider a locally constrained curvature flow in a static rotationally
symmetric space $\mathbf{N}^{n+1}$, which was firstly introduced by Hu and Li
in the hyperbolic space. We prove that if the initial hypersurface is
graphical, then the smooth solution of the flow remains to be graphical, exists
for all positive time $t\in[0,\infty)$ and converges to a slice of
$\mathbf{N}^{n+1}$ exponentially in the smooth topology. Moreover, we prove
that the flow preserves static convexity if the initial hypersurface is close
to a slice of $\mathbf{N}^{n+1}$ in the $C^1$ sense. As applications, we prove
a family of weighted geometric inequalities for static convex domains which is
close to a slice of $\mathbf{N}^{n+1}$ in the $C^1$ sense.
