  An alternate Lagrangian scheme at discrete times is proposed for the
approximation of a nonlinear continuity equation arising as a mean-field limit
of spatially inhomogeneous evolutionary games, describing the evolution of a
system of spatially distributed agents with strategies, or labels, whose payoff
depends also on the current position of the agents. The scheme is Lagrangian,
as it traces the evolution of position and labels along characteristics and is
alternate, as it consists of the following two steps: first the distribution of
strategies or labels is updated according to a best performance criterion and
then this is used by the agents to evolve their position. A general convergence
result is provided in the space of probability measures. In the special cases
of replicator-type systems and reversible Markov chains, variants of the
scheme, where the explicit step in the evolution of the labels is replaced by
an implicit one, are also considered and convergence results are provided.
