The description of the behaviour of local minima or evolution problems for families of energies cannot in general be deduced from their Gamma-limit, which is a concept designed to treat static global minimum problems. Nevertheless this can be taken as a starting point. Various issues that have been addressed are:
Find criteria that ensure the convergence of local minimizers and critical points. In case this does not occur then modify the Gamma-limit in order to match this requirement. We note that in this way we `correct' some limit theories, finding (or `validating') other ones present in the literature;
Modify the concept of local minimizer, so that it may be more `compatible' with the process of Gamma-limit;
Treat evolution problems for energies with many local minima obtained by a time-discrete scheme introducing the notion of `minimizing movements along a sequence of functionals'. In this case the minimizing movement of the Gamma-limit can always be obtained by a choice of the space- and time-scale, but more interesting behaviors can be obtained at a critical ratio between them. In many cases a `critical scale' can be computed and an effective motion, from which all other minimizing movements are obtained by scaling.
Relate minimizing movements to general variational evolution results, in particular recent theories of quasistatic motion and gradient flow in metric spaces.
I will illustrate some of these points.