I. J. Le Creurer, F. Marmolejo, E. M. Vitale, Beck’s theorem for pseudo-monads, J. Pure Appl. Algebra 173 (2002), no. 3, 293–313.
In 2-category-theory, a notion of 2-fibered categories are defined in Gray’s work and then again introduced and discussed at length by Claudio Hermida, who has good ideas on higher n (and I will be trilled to hear once that he found the time to return to the topic and give us good answers). An appendix in
Claudio Hermida, Descent on 2-fibrations and strongly 2-regular 2-categories, Appl. Categ. Structures 12 (2004), no. 5-6, 427–459.
is discussing a 2-categorical version of Beck-Chevalley condition needed to compare it with the 2-monadic descent; there is also a version of Beck’s theorem sketched there.