We study the dynamical process of equilibration of topological properties in quantum many-body systems undergoing a parameter quench between two topologically inequivalent Hamiltonians. This scenario is motivated by recent experiments on ultracold-atomic gases, where a trivial initial state is prepared before the Hamiltonian is ramped into a topological insulator phase. While the many-body wave function must stay topologically trivial in the coherent postquench dynamics, here we show how the topological properties of the single-particle density matrix dynamically change and equilibrate in the presence of interactions. In this process, the single-particle density matrix goes through a characteristic level crossing as a function of time, which plays an analogous role to the gap closing of a Hamiltonian in an equilibrium topological quantum phase transition. As an exact case study exemplifying this mechanism, we numerically solve the quench dynamics of an interacting one-dimensional topological insulator.