We investigate the magnetic-domain wall (DW) dynamics in uniaxial∕biaxial-nanowires under a thermal gradient (TG). The findings reveal that the DW propagates toward the hotter region in both nanowires. In uniaxial-nanowire, the DW propagates accompanying a rotation of the DW-plane. In biaxial nanowire, DW propagates in the hotter region, and the so-called Walker breakdown phenomenon is observed. The main physics of such DW dynamics is the magnonic angular momentum transfer to the DW. The hard (shape) anisotropy exists in biaxial-nanowire, which contributes an additional torque; hence DW speed is larger than that in uniaxial-nanowire. But rotational speed is lower initially as hard anisotropy suppresses the DW-rotation. After certain TG, DW-plane overcomes the hard anisotropy; thus, the rotational speed increases again. DW dynamics show a decreasing trend with the damping since the magnon propagation length decreases. Therefore, the above findings might be useful to realize the spintronics (i.e., fast racetrack memory) devices.