  We analyse the asymptotic behaviour of solutions of the Teichm\"uller
harmonic map flow from cylinders, and more generally of `almost minimal
cylinders', in situations where the maps satisfy a Plateau-boundary condition
for which the three-point condition degenerates. We prove that such a
degenerating boundary condition forces the domain to stretch out as a boundary
bubble forms. Our main result then establishes that for prescribed boundary
curves that satisfy Douglas' separation condition, these boundary bubbles will
not only be harmonic but will themselves be branched minimal immersions.
Together with earlier work, this in particular completes the proof that the
Teichm\"uller harmonic map flow changes every initial surface in $\mathbb{R}^n$
spanning such boundary curves into a solution of the corresponding
Douglas-Plateau problem.
