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The control of ferroelectric domain walls and their dynamics on the nanoscale becomes increasingly important for advanced nanoelectronics and novel computing schemes. One common approach to tackle this challenge is the pinning of walls by point defects. The fundamental understanding of how different defects influence the wall dynamics is, however, incomplete. In particular, the important class of defect dipoles in acceptor-doped ferroelectrics is currently underrepresented in theoretical work. In this study, we combine molecular dynamics simulations based on an ab initio derived effective Hamiltonian and methods from materials informatics and analyze the impact of these defects on the motion of 180° domain walls in tetragonal {BaTiO$_3$}⁠. We show how these defects can act as local pinning centers and restoring forces on the domain structure. Furthermore, we reveal how walls can flow around sparse defects by nucleation and growth of dipole clusters, and how pinning, roughening, and bending of walls depend on the defect distribution. Surprisingly, the interaction between acceptor dopants and walls is short-ranged. We show that the limiting factor for the nucleation processes underlying wall motion is the defect-free area in front of the wall.