Current-driven magnetic-octupole domain walls in noncollinear antiferromagnets are promising for fast spintronic devices, yet their transient dynamics are not yet fully understood. Here, we investigate spin-orbit-torque-driven domain-wall motion in Mn3Sn using a multipole micromagnetic framework. The three symmetry-related 60° walls exhibit equivalent current-driven dynamics, unlike their field-driven motion. For a representative 60° wall, the post-pulse motion is governed by relaxation of the current-induced rotation of the surrounding octupole domains, with a characteristic time of approximately 33 ps quantitatively captured by a uniform-domain analytic model. By resolving spatially nonuniform texture dynamics beyond this analytic description, the simulations further reveal an additional slow relaxation in a composite 180° wall associated with the relative motion of its constituent walls. The post-pulse response also varies systematically with the exchange stiffness and magnetic anisotropies. These results identify distinct domain and internal-coordinate relaxation channels and establish the intrinsic time scales governing pulse-controlled positioning of magnetic-octupole domain walls in antiferromagnetic spintronic devices.