  We study mean curvature flows in a warped product manifold defined by a
closed Riemannian manifold and $\mathbb{R}$. In such a warped product manifold,
we can define the notion of a graph, called a geodesic graph. We prove that the
curve shortening flow preserves a geodesic graph for any warping function, and
the mean curvature flow of hypersurfaces preserves a geodesic graph for some
monotone convex warping functions. In particular, we consider some warping
functions that go to zero at infinity, which means that the curves or
hypersurfaces go to a point at infinity along the flow. In such a case, we
prove the long-time existence of the flow and that the curvature and its
higher-order derivatives go to zero along the flow.
