We study analytically and with the numerical time-evolving block decimation method the dynamics of an impurity in a bath of spinless fermions with nearest-neighbor interactions in a one-dimensional lattice. The bath is in a Mott insulator state with alternating sites occupied and the impurity interacts with the bath by repulsive on-site interactions. We find that when the magnitudes of the on-site and nearest-neighbor interactions are close to each other, the system shows excitations of two qualitatively distinct types. For the first type, a domain wall and an antidomain wall of density propagate into opposite directions, while the impurity stays at the initial position. For the second one, the impurity is bound to the antidomain wall while the domain wall propagates, an excitation where the impurity and bath are closely coupled.