I think the point of a crossover is to explore a relationship between either two or more canons-at-large, or between a handful of characters from those canons. Therefore they will focus on points of similarity or points of dramatic conflict. Anything else isn't interesting enough to write about, generally speaking.

Part of the problem with classifying crossovers is that every canon is different, so the particular tricks needed to make Canon A merge smoothly with Canon B are not the same tricks needed to make Canon A merge smoothly with Canon C, even if you're ostensibly using the same mechanism to smush together the canons in question.