  We define a geometric flow that is designed to change surfaces of cylindrical
type spanning two disjoint boundary curves into solutions of the
Douglas-Plateau problem of finding minimal surfaces with given boundary curves.
We prove that also in this new setting and for arbitrary initial data,
solutions of the Teichm\"uller harmonic map flow exist for all times.
Furthermore, for solutions for which a three-point-condition does not
degenerate as $t\to\infty$, we show convergence along a sequence $t_i\to\infty$
to a critical point of the area given either by a minimal cylinder or by two
minimal discs spanning the given boundary curves.
