  We study continuous-time (variable speed) random walks in random environments
on $\mathbb{Z}^d$, $d\ge2$, where, at time $t$, the walk at $x$ jumps across
edge $(x,y)$ at time-dependent rate $a_t(x,y)$. The rates, which we assume
stationary and ergodic with respect to space-time shifts, are symmetric and
bounded but possibly degenerate in the sense that the total jump rate from a
vertex may vanish over finite intervals of time. We formulate conditions on the
environment under which the law of diffusively-scaled random walk path tends to
Brownian motion for almost every sample of the rates. The proofs invoke Moser
iteration to prove sublinearity of the corrector in pointwise sense; a key
additional input is a conversion of certain weighted energy norms to ordinary
ones. Our conclusions apply to random walks on dynamical bond percolation and
interacting particle systems as well as to random walks arising from the
Helffer-Sj\"ostrand representation of gradient models with certain non-strictly
convex potentials.
