  In a class of periodically driven systems, multifractal states in
non-equilibrium conditions and robustness of dynamical localization when the
driving is made aperiodic have received considerable attention. In this paper,
we explore a family of one-dimensional Aubry-Andr\'e-Harper models that are
quasi-periodically kicked with protocols following different binary
quasi-periodic sequences, which can be realized in ultracold atom systems. The
relationship between the systems' localization properties and the sequences'
mathematical features is established utilizing the Floquet theorem and the
Baker-Campbell-Hausdorff formula. We investigate the multifractality and
pre-thermalization of the eigenstates of the unitary evolution operator
combined with an analysis of the transport properties of initially localized
wave packets. We further contend that the quasi-periodically kicked
Aubry-Andr\'e-Harper model provides a rich phase diagram as the periodic case
but also brings the range of parameters to observe multifractal states and
pre-thermalization to a regime more amenable to experiments.
