  In this note we study a large class of mean curvature type flows of graphs in
product manifold $N\times R$ where N is a closed Riemann- ian manifold. Their
speeds are the mean curvature of graphs plus a prescribed function. We
establish long time existence and uniformly convergence of those flows with a
barrier condition and a condition on the derivative of prescribed function with
respect to the height. As an application we construct a weighted mean curvature
flow in large classes of warped product manifolds which evolves each graph into
a totally ge- odesic slice
