The authors analyze the smoothing property of symmetric inexact Uzawa methods as part of a multigrid method for solving symmetric saddle point problems. Additive Schwarz-type iterations are contained as a special case. As an example the theoretical results are applied to the Crouzeix-Raviart mixed finite element method for the Stokes equation. Numerical examples are presented, also for multiplicative Schwarz-type iterations that turn out to perform more efficiently as the additive version.