The analogue of the notion of Grothendieck fibration generalized from categories to multicategories.
For a multicategory regarded as a (non-symmetric) operad, discrete fibrations over it are equivalent to algebras over that operad (Hermida, proposition 5.1).
For symmetric multicategories we have the following. Let P be a symmetric operad over Set
The operadic Grothendieck construction induces an equivalence of 2-categories
between the weak algebras over P and op-fibrations over P.
This is (Heuts, theorem 1.6).
Fibrations over the terminal multicategory are equivalently representable multicategories (Hermida, corollary 4.3).
The generalization to the context of (∞,1)-operads is given by the notion of Cartesian fibration of dendroidal sets.
Fibrations of planar multicategories are discussed in
For symmetric multicategories a discussion of (op)fibrations and of the operadic Grothendieck construction is in section 1 of