  We study thermally activated, low temperature equilibrium dynamics of elastic
systems pinned by disorder using one loop functional renormalization group
(FRG). Through a series of increasingly complete approximations, we investigate
how the field theory reveals the glassy nature of the dynamics, in particular
divergent barriers and barrier distributions controling the spectrum of
relaxation times. A naive single relaxation time approximation for each
wavevector is found to be unsatisfactory. A second approximation based on a
random friction model, yields a size (L) dependent log-normal distribution of
relaxation times (mean barriers ~L^\theta and variance ~ L^{\theta/2}) and a
procedure to estimate dynamical scaling functions. Finally, we study the full
structure of the running dynamical effective action within the field theory. We
find that relaxation time distributions are non-trivial (broad but not
log-normal) and encoded in a closed hierarchy of FRG equations. A thermal
boundary layer ansatz (TBLA) appears as a consistent solution. It extends the
one discovered in the statics which was shown to embody droplet thermal
fluctuations. Although perturbative control remains a challenge, the structure
of the dynamical TBLA which encodes barrier distributions opens the way for
deeper understanding of the field theory approach to glasses.
