Consider $(X,\mathcal{A}_X)$ and $(Y,\mathcal{A}_Y)$ two Azumaya varieties over a field $k$. Recall that an Azumaya variety is the data of a variety and a sheaf of semisimple $\mathcal{O}_X$-algebra $\mathcal{A}_X$. For my purpose, we can assume $k=\mathbb{C}$ and both $X$ and $Y$ are smooth and proper.
Assume that $f : (X,\mathcal{A}_X) \to (Y,\mathcal{A}_Y)$ is a strict morphism (that is, the data of a morphism $f_0$ between $X$ and $Y$ such that $f_0^*\mathcal{A}_Y \simeq \mathcal{A}_X$).
In [Kuznetsov, "Hyperplane sections and derived categories" https://arxiv.org/pdf/math/0503700.pdf ], Kuznetsov defines (appendix D) the pushforward by $f$ of a coherent $\mathcal{A}_X$-module $F$ as the sheaf $(f_0)_*F$ with an $\mathcal{A}_Y$-module structure induced by
$$(f_0)_*F \otimes_{\mathcal{O}_Y}\mathcal{A}_Y \simeq (f_0)_*(F\otimes_{\mathcal{O}_X} \mathcal{A}_X) \to (f_0)_*F$$
Denote by $Rf_*$ the derived functor of $f_*$.
In the same paper, Kuznetsov gives the following definition (after lemma $2.1$):
Denote $p_X,p_Y$ the projection from $X\times Y$ to $X$ and $Y$ respectively. Given an object $K\in D^b(X\times Y, \mathcal{A}_X^{opp} \boxtimes \mathcal{A}_Y)$, we define the integral functor $\phi_K : D^b(X,\mathcal{A}_X) \to D^b(Y,\mathcal{A}_Y)$ with kernel $K$ as the functor
$$F \mapsto (Rp_Y)_*\left((p_X)^*_0F\otimes_{\mathcal{O}_X\boxtimes \mathcal{A}_Y}K \right)$$
Question 1 :
Is the functor $Rf_*$ isomorphic to an integral functor ? If it is, what is the kernel ?
To me the best candidate is $\mathcal{O}_{\Gamma_f}$ but I'm not sure how to endow it with a $\mathcal{A}_X^{opp}\boxtimes \mathcal{A}_Y$-module structure.
My guess is to consider the strict closed immersion morphism $j : (\Gamma_f,j_0^*(\mathcal{A}_X^{opp}\boxtimes \mathcal{A}_Y)) \to (X\times Y,\mathcal{A}_X^{opp}\boxtimes \mathcal{A}_Y)$ and consider $Rj_*\mathcal{O}_{\Gamma_f}$.
Question 2 : Given an integral functor $\phi_K : D^b(X,\mathcal{A}_X) \to D^b(Y,\mathcal{A}_Y)$ and a morphism $g :(Z,\mathcal{A}_Z) \to (X,\mathcal{A}_X)$, do we have the formula (which holds true for usual derived categories of coherent sheaves, [Huybrechts, Fourier-Mukai transform in algebraic geometry, ex. 5.12])
$$\phi_K \circ Rg_* \simeq \phi_\mathcal{R}$$
with $\mathcal{R}\simeq (g\times id_Y)^*K$.
I think the same proof as for usual $\mathcal{O}$-sheaves work if we have the expected description of $g_*$ in Question 1.
Best Answer
Your definition of a Fourier-Mukai functor is strange: $(p_X)_0^*$ doesn't have a structure of $\mathcal{A}_Y$-module, so the tensor product doesn't make sense. The definition in the paper is $$ F \mapsto (p_Y)_*((p_X)^*_0 F \otimes_{\mathcal{A}_X} K). $$
Q1: Note that $\Gamma_f \cong X$, so one can consider it with the sheaf of algebras $\mathcal{A}_X$. Then, indeed, the Fourier-Mukai functor with kernel $(\Gamma_f,\mathcal{A}_X)$.
Q2: Yes, this is true and follows from base change and projective formula, in the same way is for usual varieties.