The Bénabou-Roubaud theorem [BR70] and Beck's monadicity theorem allow us to characterize effective descent morphisms in locally cartesian closed categories; these are exactly the pullback-stable regular epimorphisms. However, characterization of effective descent morphisms is far from trivial in arbitrary categories (see, for instance, the study of continuous maps effective for descent in [RT94]).
Given a cartesian monad (T, m, e) on a category B with pullbacks, Burroni introduced a generalized notion of multicategories internal to B: these are denoted by (internal) T-multicategories [Bur71; Her00], which form a category Cat(T, B). The goal of the upcoming paper [LP] is the study of effective descent functors between internal T-multicategories.
Having described Cat(T, B) as a strict limit of a diagram D of functor categories, for which effective descent morphisms admit a characterization in terms of morphisms in B,w e mainly rely on two approaches: Lucatelli Nunes [Luc18] found sufficient conditions for effective descent morphisms in a bilimit of a diagram of categories, and then, noting that Cat(T, B) is a pullback-closed, full subcategory of the bilimit of D, we can apply a classical result (check, for instance, [JST04, Corollary 3.9]).
In this talk, we discuss some of the main ideas of this work, and give some examples of cartesian monads on suitable categories B.

References:

[BR70] Jean Bénabou and Jacques Roubaud. "Monades et descente". Comptes Rendus Hebdomadaires des Séances de L'academie des Sciences. A 270 (1970), 96-98.

[Bur71] Albert Burroni. "T-catégories (catégories dans un triple)". Cahiers de topologie et géométrie différentielle catégoriques 3.12 (1971), 215-321.

[Her00] Claudio Hermida. "Representable Multicategories". Advances in Mathematics 151 (2000), 164-225.

[JST04] George Janelidze, Manuela Sobral, and Walter Tholen. "Beyond Barr Exactness: Effective Descent Morphisms". In: Categorical Foundations: Special Topics in Order, Topology, Algebra and Sheaf Theory. Ed. by Maria Cristina Pedicchio and Walter Tholen. Encyclopedia of Mathematics and its Applications. Cambridge University Press, 2004, pp. 359-406.

[LP] Fernando Lucatelli Nunes and Rui Prezado. "Effective descent in internal multicategories". In preparation.

[Luc18] Fernando Lucatelli Nunes. "Pseudo-Kan Extensions and Descent Theory". Theory and Applications of Categories 33.15 (2018), 390-448.

[RT94] Jan Reiterman and Walter Tholen. "Effective descent maps of topological spaces". Topology and its Applications 57 (1994), 53-69.