In this paper we explore the effects of nonadiabatic external driving on the dynamics of an electronic system coupled to two electronic leads and to a phonon mode, with and without damping. In the limit of slow driving, we establish nonadiabatic corrections to thermodynamic and transport quantities. In particular, we study the first-order correction to the work done by the driving, the charge current, and the vibrational excitation using a perturbative expansion. We then compare the results to the numerically exact hierarchical equations of motion (HEOM) approach. Furthermore, the HEOM analysis spans both the weak and strong system-bath coupling regime and the slow- and fast-driving limits. We show that the electronic friction and the nonadiabatic corrections to the charge current provide a clear indicator for the Franck-Condon effect and for nonresonant tunneling processes. We also discuss the validity of the approximate quantum master equation approach and the benefits of using HEOM to study nonadiabatically driven open quantum systems out of equilibrium.