Simply stated, I've been trying for a long time to either find in the literature, or derive myself, a notion of path in Cech closure spaces, that specialises to paths in a topological space, and to graph-like paths in so-called "quasi-discrete closure spaces".

Let me recall the definitions:

A closure space is a pair $(X,C)$ where $C : \mathcal P (X) \to \mathcal P (X)$ is a function satisfying $C(\emptyset) = \emptyset$, $A \subseteq C(A)$, $C(A \cup B) = C(A) \cup C(B)$.

A continuous function $f$ is a function between two spaces such that $f(C(A)) \subseteq C(f(A))$

A topological space is (via the Kuratowski definition) a closure space with the additional axiom $C(C(A)) = C(A)$ (idempotence of closure).

Any reflexive relation $R$ generates a closure space by $C(A) = \{y \in A | \exists x \in A . x R y\}$. That's called a "quasi-discrete closure space".

Topological paths are defined as continuous functions from the unit interval.

Let me now make two examples.

Example 1: $\mathbb R^2$. Topological paths work fine (indeed!).

Example 2: the closure space on $\mathbb N$ generated by the successor relation. It's a nice closure space, but topological paths exist that do not "follow the edges" of non-symmetric relations, due to "directionality" of $R$; topology (e.g. the unit interval) is intrinsically symmetric; relations are not. For an example of this consider the set $\{a,b\}$ and the relation $R = \{ (a,b) \}$. This generates a quasi-discrete closure space. Consider the function $f : [0,1] \to \{a,b\}$ with $f(0) = b$ and $f((0,1])) = a$. This function is continuous but not a graph-like path in $R$.

Further clarifications (due to comments)

I understand that [0,1]-paths in topological spaces can't be directional. That's absolutely the case and for good reasons. But then, is there a more general construction that becomes the "natural" notion of path in closure spaces, and in topological spaces, it is not directional, since this is very natural in topological spaces?

Let's say it more formally: perhaps there's a universal construction in the category of closure spaces, of which topological spaces are a full subcategory, that captures the notion of paths in such a way that in directional graph structures like quasi-discrete closure spaces, paths are directional, and in topological spaces, paths are in one-to-one-correspondence to classical, topological paths, a.k.a. $[0,1]$-morphisms?