Firstly, you could be discrete without being a lattice, for instance you could make a bunch of different shaped tetrahedra such that each region of space was inside a tetrahedra. If the tetrahedra were randomly sheared but all had a kind of regularity where they were all big enough to contain a sphere of radius A and be contained in a sphere of radius B, then I think the best distance along edges might approach a fixed multiple of the true distance as the continuous crow flies. And I mention this because this is a not uncommon way to discretize space.
However, you basically asked not just for a discrete space, but for a lattice space. And if you truly want that have that at every time you are located on the corner of some cube that is nicely packed inside a larger cube with identical cubes and that larger cube is itself nicely packed inside a still larger cube with identical cubes and so on. Then it is fairly easy to get some physics similar to the real world by allowing things to jump from neighbor to neighbor at different speeds.
For instance we don't know where particles are located exactly. We can measure them accurately enough to say they are in some box at one time, and then in a possibly different box at a later time. And we can say that they must have had a certain average velocity to get from here the there in that time interval.
And based on the velocity (where you ended up relative to where you started all divided by how long it took you to get from here to there) you can infer the distance travelled. But you can't actually see them perfectly so you don't know how far they actually traveled. You can use fancy equipment and place them in very small boxes but you still don't know where in the box they are or how they are moving, just that at one time they are in this box and then at another time they are in that box.
That is exactly what reporting position to finite accuracy means. So you really effectively have boxes based on how well you are measuring things. So what if you imagined smaller boxes inside those and yet smaller boxes inside those. Are the particles moving faster along the x and y axis going from corner to corner of those super small boxes inside the small boxes inside the actual boxes that you can measure? Who knows (or cares) what is happening when you aren't looking.
No one thinks they are doing that because we would be surprised if someday we started measuring things to better accuracy and all of a sudden things behaved differently and have preferred directions, in fact not having preferred direction is part of how we predict what to expect.
But even if we don't expect it, it could happen because we don't know what is going on faster than we look and in regions smaller than we can look inside.
So if you get scientific about distance travelled you see that really it is about being in one place at a time and being at another place at a different time and us tracking what you did as much as we felt like and then saying that more accurately that we just don't know.
When you don't know what the particle did you can base your answer on the geometry of the boxes instead. You literally imagine every line that started in one box and ended on the other box and take the average. Or you imagine every possible path from one vox to the other and take minimum (this will underestimate distance travelled but the error goes away as the boxes get smaller, it's like rounding down to the nearest \$1000 then rounding down to the nearest \$100 then nearest \$10 then nearest \$1 you are always underpaying but the error gets smaller, and unlike money we could make the boxes as small as we want).
So you can do the same. To go at a certain velocity just take certain ratios of steps left versus forward and take them faster based on the overall velocity. But then rethink distance not as the unknowable how do things actually move but instead on what is the geometrical relationship between where you started and stopped to someone that can't see exactly how you move.
Is this related to actual models physicists make? Not really. lattice theories in Physics are usually for fields that have values at every single lattice point and you are computing how they change, so it is already everywhere and you just want to know how it is changing.
Like if you have temperature, you could imagine a value of temperature at all the lattice points, and then if you have a hot point surrounded by cooler points you could have it get colder and have an equal amount of heat flow to the nearby points.
That's something that happens on some physicists models, but really just because the other models were too hard.
If you just have some particles it is easier yo just say where they are rather than have a bunch of points most of which are empty.