  We consider inverse curvature flows in warped product manifolds, which are
constrained subject to local terms of lower order, namely the radial coordinate
and the generalized support function. Under various assumptions we prove
longtime existence and smooth convergence to a coordinate slice. We apply this
result to deduce a new Minkowski type inequality in the anti-de-Sitter
Schwarzschild manifolds and a weighted isoperimetric type inequality in the
hyperbolic space.
