We study the dynamics of topological bound states following their coupling to the Fermi sea of a topologically trivial, gapless lead. Specifically, we study the quench dynamics of solitons in the Su-Schrieffer-Heeger model and of Majorana zero modes of Kitaev model. Remarkably, we find bound states propagate through the lead preserving their topological nature, such as fractional charge and exchange statistics. We explain this phenomenon as a manifestation of dynamically induced topology in the Fermi sea arising from the entan- glement between the topological and gapless systems. We obtain, both analytically and numerically, topological features of propagating bound states, including their fractional charge, charge fluctuations, entanglement entropy, and fractional statistics. This allows us to characterize the coherence time over which these topological features relax. We also study the effects of interactions and disorder in the lead on the integrity of the topological bound states and the coherence time of dynamically induced topology. Supported by ISF grants 401/12 and 1626/16, the EU Seventh Framework Programme (FP7/2007-2013) grant 303742, BSF grant 2014345, NSF grants PHY-1607611 and DMR-1350663 and H.B.C. fellowship (Israel).