  Local observables in generic periodically driven closed quantum systems are
known to relax to values described by periodic infinite temperature ensembles.
At the same time, ergodic static systems exhibit anomalous thermalization of
local observables and satisfy a modified version of the eigenstate
thermalization hypothesis (ETH), when disorder is present. This raises the
question, how does the introduction of disorder affect relaxation in
periodically driven systems? In this work, we analyze this problem by
numerically studying transport and thermalization in an archetypal example. We
find that thermalization is anomalous and is accompanied by subdiffusive
transport with a disorder dependent dynamical exponent. Distributions of matrix
elements of local operators in the eigenbases of a family of effective
time-independent Hamiltonians, which describe the stroboscopic dynamics of such
systems, show anomalous departures from predictions of ETH signaling that only
a modified version of ETH is satisfied. The dynamical exponent is shown to be
related to the scaling of the variance of these distributions with system size.
