Is there any relation between Lieb-Robinson velocity bounds and micro-causality?

Background

So I recently asked a question about relativistic quantum mechanics and the answerer invoked micro-causality (from QFT) to show me that the assumption the information would propagate infinitely fast due to the sudden approximation was false. After talking to a friend working in quantum information theory he said that even in regular quantum mechanics there are Lieb-Robinson velocity bounds:

The existence of such a finite speed was discovered mathematically by Lieb and Robinson, (1972). It turns the locality properties of physical systems into the existence of, and upper bound for this speed.

When I compare this to locality statement from QFT (microcausality):

A basic characteristic of physics in the context of special relativity and general relativity is that causal influences on a Lorentzian manifold spacetime propagate in timelike or lightlike directions but not spacelike.

The fact that any two spacelike-separated regions of spacetime thus behave like independent subsystems is called causal locality … Microcausality condition a requirement that the causality condition (which states that cause must precede effect) be satisfied down to an arbitrarily small distance and time interval.

Question

Since both are statements of locality I’m curious if there is any relation between the two? Like would change into the other if you take some kind of limit as in the comment section?

… I would guess that the Lieb-Robinson bound in lattice QFT does become
the usual causality property in the continuum limit. We could probably
study this in a simple exactly-solvable example, like the lattice QFT
of a free scalar field. …