  We theoretically study the interaction of spin waves and a domain wall in a
quasi-one-dimensional easy-cone ferromagnet. The gapless spin waves on top of a
domain wall are found to exhibit finite reflection in contrast to the
well-known perfect transmission of gapful spin waves in easy-axis magnets.
Based on the obtained scattering properties, we study the thermal-magnon-driven
dynamics of a domain wall subjected to a thermal bias within the
Landau-B\"uttiker formalism, where transmitted magnons are shown to exert the
magnonic torque on the domain wall and thereby drive it with the velocity
linear to the applied thermal bias. The peculiar gapless nature of spin waves
in easy-cone magnets enables the thermally-driven domain-wall motion even at
low temperatures, differing from the easy-axis case where the domain-wall
velocity is exponentially suppressed at low temperatures. Our work suggests
that easy-cone magnets can serve as a useful platform to study the interaction
of gapless spin waves and nonlinear excitations and thereby realize
low-temperature magnon-related phenomena.
