Jacob Lurie defined a model structure on the category of marked simplicial sets sliced over a fixed simplicial set $S$ called the cartesian model structure. (For a definition, see here or HTT Ch.3.1). How does this model structure behave with respect to slicing over objects? That is, is the natural model structure on the slice over a simplicial set $S$ the same as the cartesian model structure on the slice over $S$?
Do any of the other flexible models of higher categories (specifically complete segal spaces and segal categories) have relative forms? If so, do these model categories give the right results under slicing? Whether they do or not, are there appropriate analogues of the straightening and unstraightening constructions? If so, do they give a powerful enough theory of fibered and cofibered categories (which are extremely important in ordinary 2-category theory (these give the "right" slice bicategories)).