The only natural condition I can think of is Berry's "I condition" (, Sect. 12.3):
(I) each compact element dominates finitely many elements.
The above condition is the defining property of Berry's dI-domains, which are distributive (that's what the "d" stands for) algebraic domains satisfying condition I. This is a widely known and well studied class of domains, which served as the basis for what is known as "stable" semantics (an attempt to capture the sequential behavior of the $\lambda$-calculus). Girard's coherence spaces  give a particularly simple and useful example of dI-domains. A coherence space may be seen as the set of cliques of a reflexive graph, ordered by inclusion. The compact elements are the finite cliques, which obviously satisfy your requirement.
I don't know if generalizations of the I condition have been studied. A counterexample to your property is given by the ordinal $\omega+2$ ($\omega+1$ is compact but $\omega$ is not), so certainly generalizing "finite" to "well-founded+no infinite antichains" does not work.
 Roberto Amadio and Pierre-Louis Curien. Domains and Lambda-Calculi. Cambridge Tracts in Theoretical Computer Science, 1998.
 Jean-Yves Girard, Yves Lafont and Paul Taylor. Proofs and Types. Cambridge Tracts in Theoretical Computer Science, 1989.