There are those who pursue this in one direction or another, and often they come from the same ultrafinitist leanings of Doron. Also, there are approaches, like Loop Quantum Gravity, that end up constructing a discrete space-time through more fundamental ontology. And there are related formalisms in more standard theories, like through the calculation of holographic information content from the Holographic Principle of quantum gravity, which also arrive at finite information content.

Examples of the more "brute" ways this has been pursued (read as: more from the philosophical bent but with little mathematical formalism) can be seen in papers such as:

To the finite information content of the physically existing reality

Computational capacity of the universe

Clearly, these do not provide much more than the argument of finite content, which is Zeilberger's jump point as well. But there has been definite work in the direction of discrete geometric reasoning as well. Kustaanheimo's work in the middle of last century is often taken as the starting ground here. The standard reference is:

Kustaanheimo, P., 1951, ‘A Note on a Finite Approximation of the Euclidean Plane Geometry’, Societas Scientiarum Fennica. Commentationes Physico-Mathematicae, 15 (19): 1-11.

This has led to a body of work by others who have continued in this vein. A nice summary is found at the:

Stanford Encyclopedia of Philosophy entry on Finitism in Geometry

Note that when you look at these approaches, they do not always (or even often) strictly turn limit dynamics to difference dynamics. There are some approaches where this is what occurs, and several approaches are possible that do not impose lattices or large-scale symmetry breaking, but they usually require ontologies that more radically break from the classical.

Finally, it should be pointed out that although these approaches are often pursued by those with ultrafinitist leanings, there are a large number of contributers who do not have such foundational positions.