This is a fundamental problem in phonological theory which has no trivial and/or universally accepted solution, and in the case of dissimilations there aren't even widely-accepted trendy solutions.
Here is a baseline solution, in SPE theory, using a general scheme for segment skipping:
That is, "r" becomes a lateral when preceded by "r" which may have any number of consonants and vowels intervening. SPE posited a feature "segment", which would allow the rule to state the "skipping over" part with "[+seg]₀", though pretty much everybody besides SPE rejected "segment" as a phonetic features. A fatal flaw for Georgian is that you can't actually skip all segments: you can't skip a lateral. However, that can be fixed by saying that the intervening segments are [-lat]₀.
A vast literature arose after 1968 which attempted to address the problem of expressing "intervening material", and to this day this is an unresolved and I believe unresolvable matter. While it is generally acknowledged that the stuff that can be skipped is not "any arbitrary pattern", objections have been raised to every solution. The most robust solutions, however, focus on the fact that the intervening segments, the ones skipped, are not the ones that undergo or trigger the rule. That is, if a rule say "a→b/c_", then you can't skip over a or c.
While not a complete account of the problem, autosegmental representations do the best job of capturing the facts, because in that theory, features aren't in a 1-to-1 relation with segments as they are in SPE theory. The typical autosegmental account then says that the rule involves some feature on the target and trigger segments, and the feature specifications have to be adjacent on their tier – but the segments don't have to be adjacent. So if you have a nasal dissimilation rule that refers to two nasals without consideration for whether the segments themselves are strictly adjacent, the rule just mention "nasal", and oral segments aren't specified as "nasal".
In the case of Georgian and other liquid dissimilations, the added complication is a need to refer to "liquids", not easy to do in SPE theory. However, some feature theories countenance a feature "liquid", and in those theories, the difference between [r] and [l] would be that one has a feature such as "rhotic" that the other lacks. In words, "when there are two rhotics, the rhotic feature of the second deletes" (and a liquid that is not rhotic is, in Georgian, a phonetic lateral). In the case of kartluri, the relevant substructure is:
[... r t l u r...]
Liq Liq Liq
the rule requires:
so the intervening Liquid (the /l/) which isn't a trigger blocks the rule from applying.