  We study the motion of discrete interfaces driven by ferromagnetic
interactions in a two-dimensional low-contrast periodic environment, by
coupling the minimizing movements approach by Almgren, Taylor and Wang and a
discrete-to-continuum analysis. As in a recent paper by Braides and Scilla
dealing with high-contrast periodic media, we give an example showing that in
general the effective motion does not depend only on the Gamma-limit, but also
on geometrical features that are not detected in the static description. We
show that there exists a critical value $\widetilde{\delta}$ of the contrast
parameter $\delta$ above which the discrete motion is constrained and coincides
with the high-contrast case. If $\delta<\widetilde{\delta}$ we have a new
pinning threshold and a new effective velocity both depending on $\delta$. We
also consider the case of non-uniform inclusions distributed into periodic
uniform layers.
