Periodic driving has recently attracted intense interest as a means for controlling and investigating a variety of intriguing quantum many-body phenomena. In particular, driving has been proposed as a new route for altering the topological properties of Bloch bands. In this talk, I will present our study of the transient dynamics in related periodically driven systems, focusing on the role of interactions. I will describe the appearance and time-dependent renormalization of the effective (Floquet) band gap. In addition, I will discuss the time scales for the emergence of the Floquet-Bloch bands in the non-equilibrium correlation functions, accessible in ARPES experiments.