  We investigate the spectral and transport properties of many-body quantum
systems with conserved charges and kinetic constraints. Using random unitary
circuits, we compute ensemble-averaged spectral form factors and
linear-response correlation functions, and find that their characteristic time
scales are given by the inverse gap of an effective Hamiltonian$-$or
equivalently, a transfer matrix describing a classical Markov process. Our
approach allows us to connect directly the Thouless time, $t_{\text{Th}}$,
determined by the spectral form factor, to transport properties and linear
response correlators. Using tensor network methods, we determine the dynamical
exponent, $z$, for a number of constrained, conserving models. We find
universality classes with diffusive, subdiffusive, quasilocalized, and
localized dynamics, depending on the severity of the constraints. In
particular, we show that quantum systems with 'Fredkin constraints' exhibit
anomalous transport with dynamical exponent $z \simeq 8/3$.
