  We consider first order variational MFG in the whole space, with aggregative
interactions and density constraints, such that the stationary states of the
game are contained in two isolated compact sets of mass distributions with
finite quadratic moments. Under general assumptions on the interaction
potential, we provide a method for the construction of periodic in time
solutions for the MFG, which oscillate among the two sets of static equilibria.
Moreover, as the period increases to infinity, we show that these periodic
solutions converge, in a suitable sense, to heteroclinic connections. As a
model example, we consider a MFG system where the interactions are modeled via
a Riesz-type aggregative potential with spatial preferences.
