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The Teichm\"uller harmonic map flow is a gradient flow for the harmonic mapenergy of maps from a closed surface to a general closed Riemannian targetmanifold of any dimension, where both the map and the domain metric are allowedto evolve. Given a weak solution of the flow that exists for all time $t\geq0$, we find a sequence of times $t_i\to\infty$ at which the flow at differentscales converges to a collection of branched minimal immersions with no loss ofenergy. We do this by developing a compactness theory, establishing no loss ofenergy, for sequences of almost-minimal maps.  Moreover, we construct an example of a smooth flow for which the image of thelimit branched minimal immersions is disconnected. In general, we show that thenecks connecting the images of the branched minimal immersions becomearbitrarily thin as $i\to\infty$.

Refined asymptotics of the Teichm\"uller harmonic map flow into general targets