Why, exactly, does reflexivity depend on the domain of the relation?

Question

For many properties of binary relations, like symmetry, asymmetry, and transitivity, it does not depend on what domain of the relation we are considering. More precisely, given a binary relation $R$ on $A$ and a superset $B$ of $A$, $R$ is symmetric/asymmetric/transitive on $A$ iff it is so on $B$. However, reflexivity does depend on the domain. Why? Also, bonus question, is there a logical characterization of those properties which does not depend on the domain? That is, given a formula $F$ of first-order logic, what syntactic property must it have so that $R$ satisfies $F$ in a domain-independent way?