  With the aim to understand the role of the constraints in the thermalisation
of quantum systems, we study the dynamics of a family of kinetically
constrained models arising through duality from the XXZ spin chain. We find
that integrable and nonintegrable deformations around the stochastic point give
rise to ground state phase transitions between localised and delocalised
phases, which in turn determine the nature of the relaxation dynamics at finite
energy densities. While in the delocalised phase thermalisation is fast and
homogeneous, in the localised phase relaxation is slow, temporal
autocorrelations exhibit plateaus indicative of metastability, and the growth
of entanglement is heterogeneous in space. Furthermore, by considering
relaxation from initial product states, we demostrate that this slow
thermalisation can be rationalised directly from the presence of constraints in
the dynamics.
