  Floquet theory is an indispensable tool for analysing periodically-driven
quantum many-body systems. Although it does not universally extend to classical
systems, some of its methodologies can be adopted in the presence of
well-separated timescales. Here we use these tools to investigate the
stroboscopic behaviours of a classical spin chain that is driven by a periodic
magnetic field and coupled to a thermal reservoir. We detail and expand our
previous work: we investigate the significance of higher-order corrections to
the classical Floquet-Magnus expansion in both the high- and low-frequency
regimes; explicitly probe the evolution the dynamics of the reservoir; and
further explore how the driven system synchronises with the applied field at
low frequencies. In line with our earlier results, we find that the
high-frequency regime is characterised by a local Floquet-Gibbs ensemble with
the reservoir acting as a nearly-reversible heatsink. At low frequencies, the
driven system rapidly enters a synchronised state, which can only be fully
described in a global picture accounting for the concurrent relaxation of the
reservoir in a fictitious magnetic field arising from the drive. We highlight
how the evolving nature of the reservoir may still be incorporated in a local
picture by introducing an effective temperature. Finally, we argue that
dissipative equations of motion for periodically-driven many-body systems, at
least at intermediate frequencies, must generically be non-Markovian.
