[mikhailfranco]

I think it is easy to get around the problems of Lorentz invariance in discrete models, as long as it is not the spacetime that is explicitly discretized on the lattice. The lattice must be some other combinatorial graph structured algebra, with non-local 'propagation' of fields. Quantum Mechanics says that space is only defined relationally on the intervals between field interactions: a 'particle' is only localized as a particle when it interacts (position or momentum observable). So discrete space and time appear as values on some subset of lattice nodes in response to some propagating fields ('particles' only at the interaction event). The slogan for this is 'spooky distance at an action', because it is the (inter)action that defines the space(time).

QM (and QFT) assume a background time, it is not an observable, even though it appears to commute with energy (e.g. energy is momentum in the time direction). So it's more tricky to understand how time emerges in a discrete Quantum Gravity, but I suspect there is an intrinsic proper time, defined by interactions with the 'vacuum' (minimal field states on the underlying lattice), which bootstraps a relational time defined over intervals between interactions.