  It is well understood that many-body systems driven at high frequency heat up
only exponentially slowly and exhibit a long prethermalization regime. We prove
rigorously that a certain relevant class of systems heat very slowly under weak
periodic driving at intermediate frequency as well. This class of systems are
those whose time-dependent, possibly translation-invariant, Hamiltonian is a
weak perturbation of a sum of mutually commuting terms. This condition covers
several periodically kicked systems that have been considered in the literature
recently, in particular kicked Ising models. In contrast to the high-frequency
regime, the prethermalization dynamics of our systems is in general not related
to any time-independent effective Floquet Hamiltonian. Our results also have
non-trivial implications for closed (time-independent) systems. We use the
example of an Ising model with transversal and longitudinal field to show how
they imply confinement of excitations. More generally, they show how "glassy"
kinetically constrained models emerge naturally from simple many-body
Hamiltonians, thus connecting to the topic of 'translation-invariant
localization'.
