Model paths by regular languages
If the number of locations (e.g. points or regions) are finite, naively, you can say that these locations are my states and you can directly use a DFA with an alphabet containing UP, DOWN. But you already said it's impracticable for your case.

Then, let's look into where the number of locations is infinite or simply too many. Basically, in this case, you need to recognize that the sequences UP-DOWN, UP-UP-DOWN-DOWN, UP-UP-UP-DOWN-DOWN-DOWN ... etc. are equivalent because equal number of UP and DOWNs gets you back to the starting point. This is the classical example of non-regular languages. Therefore I suggest you looking into other automata such as counter machines if they are sufficient to capture your intentions.