  We consider the parabolic polyharmonic diffusion and $L^2$-gradient flows of
the $m$-th arclength derivative of curvature for regular closed curves evolving
with generalised Neumann boundary conditions. In the polyharmonic case, we
prove that if the curvature of the initial curve is small in $L^2$, then the
evolving curve converges exponentially in the $C^\infty$ topology to a straight
horizontal line segment. The same behaviour is shown for the $L^2$-gradient
flow provided the energy of the initial curve is sufficiently small. In each
case the smallness conditions depend only on $m$.
