  The Teichm\"uller harmonic map flow is a gradient flow for the harmonic map
energy of maps from a closed surface to a general closed Riemannian target
manifold of any dimension, where both the map and the domain metric are allowed
to evolve. Given a weak solution of the flow that exists for all time $t\geq
0$, we find a sequence of times $t_i\to\infty$ at which the flow at different
scales converges to a collection of branched minimal immersions with no loss of
energy. We do this by developing a compactness theory, establishing no loss of
energy, for sequences of almost-minimal maps.
  Moreover, we construct an example of a smooth flow for which the image of the
limit branched minimal immersions is disconnected. In general, we show that the
necks connecting the images of the branched minimal immersions become
arbitrarily thin as $i\to\infty$.
