We outline results regarding two linear semi-discrete analogues of polyharmonic curve diffusion for closed polygons in ℝp, p ≥ 2, considered in recent works by the authors, [12] and [13]. In particular, fruitful discussions regarding self-similar solutions in higher codimension were held between the first author and Prof Ben Andrews at the MATRIX institute. We consider both parabolic and hyperbolic evolutions including self-similar solutions under each flow. Parabolic evolution only has non-trivial scaling self-similar solutions, whereas in the undamped hyperbolic case, self-similar solutions by pure rotation exist. Polygons evolving by the parabolic flow converge exponentially to a point, where under appropriate rescaling the limiting shape is an affine transformation of a regular polygon. This is also true in some cases for damped hyperbolic motion of polygons dependent on the damping term. Otherwise we also get oscillating motion under this flow. The hyperbolic flow also allows for prescription of an intermediate polygon state at a distinct time. As an application, we set up a semi-discrete Yau difference flow for both the parabolic and hyperbolic cases where an initial polygon converges to a target polygon in infinite time. Hyperbolic evolution further allow for an intermediate state to be obtained at a distinct time during this process.