  We introduce the abstract setting of presheaf category on a thick category of
cubes. Precubical sets, symmetric transverse sets, symmetric precubical sets
and the new category of (non-symmetric) transverse sets are examples of this
structure. All these presheaf categories share the same metric and homotopical
properties from a directed homotopy point of view. This enables us to extend
Raussen's notion of natural $d$-path for each of them. Finally, we adapt
Ziemia\'{n}ski's notion of cube chain to this abstract setting and we prove
that it has the expected behavior on precubical sets. As an application, we
verify that the formalization of the parallel composition with synchronization
of process algebra using the coskeleton functor of the category of symmetric
transverse sets has a category of cube chains with the correct homotopy type.
