  We consider the question of whether solutions of variants of Teichm\"uller
harmonic map flow from surfaces $M$ to general targets can degenerate in finite
time. For the original flow from closed surfaces of genus at least $2$, as well
as the flow from cylinders, we prove that such a finite-time degeneration must
occur in situations where the image of thin collars is `stretching out' at a
rate of at least $\text{inj}(M,g)^{-(\frac14+\delta)}$, and we construct
targets in which the flow from cylinders must indeed degenerate in finite time.
For the rescaled Teichm\"uller harmonic map flow, the condition that the image
stretches out is not only sufficient but also necessary and we prove the
following sharp result: Solutions of the rescaled flow cannot degenerate in
finite time if the image stretches out at a rate of no more than
$\lvert\log(\text{inj}(M,g))\rvert^{\frac12}$, but must degenerate in finite
time if it stretches out at a rate of at least
$\lvert\log(\text{inj}(M,g))\rvert^{\frac12+\delta}$ for some $\delta>0$.
