  Minimising movements are investigated for an energy which is the
superposition of a convex functional and fast small oscillations. Thus a
minimising movement scheme involves a temporal parameter $\tau$ and a spatial
parameter $\epsilon$, with $\tau$ describing the time step and the frequency of
the oscillations being proportional to $\frac 1 \epsilon$. The extreme cases of
fast time scales $\tau << \epsilon$ and slow time scales $\epsilon << \tau$
have been investigated in Braides, Springer Lecture Notes 2094 (2014). In this
article, the intermediate (critical) case of finite ratio $\epsilon/\tau>0$ is
studied. It is shown that a pinning threshold exists, with initial data below
the threshold being a fixed point of the dynamics. A characterisation of the
pinning threshold is given. For initial data above the pinning threshold, the
equation and velocity describing the homogenised motion are determined.
