  We numerically study the dynamics on the ergodic side of the many-body
localization transition in a periodically driven Floquet model with no global
conservation laws. We describe and employ a numerical technique based on the
fast Walsh-Hadamard transform that allows us to perform an exact time evolution
for large systems and long times. As in models with conserved quantities (e.g.,
energy and/or particle number) we observe a slowing down of the dynamics as the
transition into the many-body localized phase is approached. More specifically,
our data is consistent with a subballistic spread of entanglement and a
stretched-exponential decay of an autocorrelation function, with their
associated exponents reflecting slow dynamics near the transition for a fixed
system size. However, with access to larger system sizes, we observe a clear
flow of the exponents towards faster dynamics and can not rule out that the
slow dynamics is a finite-size effect. Furthermore, we observe examples of
non-monotonic dependence of the exponents with time, with dynamics initially
slowing down but accelerating again at even larger times, consistent with the
slow dynamics being a crossover phenomena with a localized critical point.
