  In this paper, we study families of immersed curves
$\gamma:(-1,1)\times[0,T)\rightarrow\mathbb{R}^2$ with free boundary supported
on parallel lines $\{\eta_1, \eta_2\}:\mathbb{R}\rightarrow\mathbb{R}^2$
evolving by the curve diffusion flow and the curve straightening flow. The
evolving curves are orthogonal to the boundary and satisfy a no-flux condition.
We give estimates and monotonicity on the normalised oscillation of curvature,
yielding global results for the flows.
