The general setting of this chapter involves a map defined on a pseudometric space equipped with a preorder. For such a map, we consider certain invariants which measure the deviation to a given sublevel set. The considered invariants are global (a distance to the sublevel set, or the absolute variation of the map) or local (an adequate notion of strong slope). Specifically, we study the interaction between these invariants by means of conditions which are attached to the various pairs of invariants and express a good asymptotic behaviour of the map. Our main results are implications between the conditions for the various pairs, and we prove them under suitable assumptions on the map: submonotonicity and/or an extension of convexity to the present setting of preordered pseudometric space.