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Hyland's Elements of a theory of algebraic theories describes a precise connection between multicategories and $\mathbf{Prof}$ in Section 4.3. I shall briefly describe the intuition; a complete pictur …

looking for a copy of Michel Thiébaud's 1971 thesis Self-Dual Structure-Semantics and Algebraic Categories, which appears to be an early reference for the relationship between the Kleisli construction and profunctors …

This is exactly the subject of the paper Coends of higher arity by Loregian and de Oliveira Santos.

In Bénabou's Les distributeurs, in which the bicategory of profunctors is introduced, Bénabou remarks (page 17, quoted below) that $\mathbf{Prof}$ may be viewed as the construction of a bicategory from … (Indeed, for every functor $F : A \to B$ between small categories, we have profunctors $F_* : A \nrightarrow B$ and $F^* : B \nrightarrow A$ given by $F_*(b, a) = B(b, Fa)$ and $F^*(a, b) = B(Fa, b)$, …

I discovered a related characterisation in Betti's Formal theory of internal categories. For $\mathcal E$ a finitely complete category, Betti claims (in the theorem at the top of page 49) that $\mathb …

I shall sketch out a proof that $\mathbf{Prof}$ is almost obtained from $\mathbf{Cat}$ by adjoining right adjoints to every 1-cell, following Roald Koudenburg's suggestions in the comments. The remain …

Let $P : W° \times Y \to \mathbf{Set}$ and $Q : X° \times V \to \mathbf{Set}$ be profunctors, and let $L : X \to W$ and $R : Y \to V$ be functors. … In particular, it would be useful to prove statements about these "adjunctions with respect to profunctors", for instance giving characterisations of (co)reflective adjunctions with respect to profunctors …

It is well known that a distributor/profunctor $A \not\rightarrow B$, i.e. a functor $B^{\text{op}} \times A \to \mathrm{Set}$, is equivalent to a two-sided discrete fibration from $A$ to $B$. Further …

Let $P$ be a distributor/profunctor from a small category $A$ to a small category $B$, i.e. a functor $P : B^\circ \times A \to \mathrm{Set}$. We may then define a distributor from $[A^\circ, \mathrm{ …

I think it worth mentioning that precisely the notion described in the question is given in Cockett–Koslowski–Seely's Morphisms and modules for poly-bicategories, where it is called a multi-bicategory …