  Motivated by pedestrian modelling, we study evolution of measures in the
Wasserstein space. In particular, we consider the Cauchy problem for a
transport equation, where the velocity field depends on the measure itself.
  We deal with numerical schemes for this problem and prove convergence of a
Lagrangian scheme to the solution, when the discretization parameters approach
zero. We also prove convergence of an Eulerian scheme, under more strict
hypotheses. Both schemes are discretizations of the push-forward formula
defined by the transport equation. As a by-product, we obtain existence and
uniqueness of the solution.
  All the results of convergence are proved with respect to the Wasserstein
distance. We also show that $L^1$ spaces are not natural for such equations,
since we lose uniqueness of the solution.
