  We study how stable the Majorana-mediated spin transport in a quantum spin
Kitaev model is against thermal fluctuations. Using the time-dependent thermal
pure quantum state method, we examine finite-temperature spin dynamics in the
Kitaev model. The model exhibits two characteristic temperatures $T_L$ and
$T_H$, which correspond to energy scales of the local flux and the itinerant
Majorana fermion, respectively. At low temperatures $(T\ll T_L)$, an almost
flux-free state is realized and the spin excitation propagates in a similar way
to that for the ground state. Namely, after the magnetic pulse is introduced at
one of the edges, the itinerant Majorana fermions propagate the spin
excitations even through the quantum spin liquid state region, and oscillations
in the spin moment appear in the other edge with a tiny magnetic field. When
$T\sim T_L$, larger oscillations in the spin moments are induced in the other
edge, compared to the results at the ground state. At higher temperatures,
excited $Z_2$ fluxes disturb the coherent motion of the itinerant Majorana
fermions, which suppresses the spin propagation. Our results demonstrate a
crucial role of thermal fluctuations in the Majorana-mediated spin transport.
