The so-called fiber lay-down models arise in the production process of nonwovens. We introduce the generalized version of the basic fiber lay-down model which can precisely be formulated in abstract form as some manifold-valued stochastic differential equation. An important criterion for the quality of the nonwoven material is how the solution to the associated Fokker-Planck equation converges towards its stationary state. Especially, one is interested in determining the speed of convergence. Here we present some results concerning the long-time behavior by using classical stochastic methods as well as modern analytic methods from the theory of hypocoercivity. Demanding mathematical difficulties arising since the equation is degenerate. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)