  Extended reservoirs provide a framework for capturing macroscopic, continuum
environments, such as metallic electrodes driving a current through a nanoscale
contact, impurity, or material. We examine the application of this approach to
periodically driven systems, specifically in the context of quantum transport.
As with non--equilibrium steady states in time--independent scenarios, the
current displays a Kramers' turnover including the formation of a plateau
region that captures the physical, continuum limit response. We demonstrate
that a simple stability criteria identifies an appropriate relaxation rate to
target this physical plateau. Using this approach, we study quantum transport
through a periodically driven tilted lattice coupled to two metallic reservoirs
held at a finite bias and temperature. We use this model to benchmark the
extended reservoir approach and assess the stability criteria. The approach
recovers well--understood physical behavior in the limit of weak
system--reservoir coupling. Extended reservoirs enable addressing strong
coupling and non--linear response as well, where we analyze how transport
responds to the dynamics inside the driven lattice. These results set the
foundations for the use of extended reservoir approach for periodically driven,
quantum systems, such as many--body Floquet states.
