We use drifted Brownian motion in tailor made warped product model spaces as comparison constructions to sudy $p$-hyperbolicity and $p$-parabolicity of a large class of submanifolds in ambient spaces with restricted curvature behaviour. It is expected that similar techniques will also give generalized isoperimetric inequalities as well as $p$-Laplace heat kernel estimates and mean exit time estimates for the corresponding nonlinear diffusion.