I can't speak for these authors, but what I understand by a "Fourier-Mukai" transform between Fukaya categories is *the functor between extended Fukaya categories associated with a Lagrangian correspondence*. I expect these will appear a good deal in the next few years, in symplectic topology and its applications to low-dim topology and mirror symmetry (cf. these papers of Auroux and Abouzaid-Smith).

A *Lagrangian correspondence* from $X$ to $Y$ is an embedded Lagrangian submanifold of $X\times Y$ with symplectic form $(-\omega_X)\oplus\omega_Y$. One can take the graph of a symplectomorphism, for instance, or the vanishing moment-map locus $\mu^{-1}(0)$ as a correspondence from a Hamiltonian $G$-manifold $M$ to the quotient $\mu^{-1}(0)/G$. According to Wehrheim-Woodward, a *generalised Lagrangian* in $X$ is a sequence of symplectic manifolds $X_0=pt., X_1, X_2,\dots,X_d=X$ and Lagrangian correspondences $L_{i,i+1}$ from $X_i$ to $X_{i+1}$. Generalised Lagrangians (subject to the usual sorts of restrictions and decorations) form objects in the *extended Fukaya category* $F^{\sharp}(X)$, whose $A_\infty$-structure is under construction by Ma'u-Wehrheim-Woodward. If it happens that two adjacent Lagrangian correspondences have a smooth, embedded composition, say $L' = L_{i+1,i+2}\circ L_{i,i+1}$, then deleting $X_{i+1}$ from the chain and substituting $L'$ for its two factors results in an isomorphic object; see arxiv:0905.1368.

The geometric mechanism behind this is the idea of "pseudo-holomorphic quilts", see arXiv:0905.1369, arxiv:math.SG.0606061.

Whilst $F^\sharp(X)$ seems even less tractable than the Fukaya category $F(X)$, we expect that in many cases the embedding $F(X)\to F^\sharp(X)$ induces a quasi-isomorphism of module categories.
A Lagrangian correspondence from $X$ to $Y$ induces a "Fourier-Mukai" $A_\infty$-functor between extended Fukaya categories. Even better, we expect that in this way one gets an $A_\infty$-functor $F(X_-\times Y)\to \hom(F^{\sharp}(X),F^{\sharp}(Y))$.

*Interpolated paragraph, in response to Kevin's query*: The definition of the $A_\infty$-functor associated with a correspondence $C$ from $X$ to $Y$ is very simple, at least on objects. An object of $F^\sharp(X)$ is a chain of Lagrangian correspondences, beginning at the 1-point manifold and ending at $X$. We just tack $C$ to the end of this sequence. The clever thing about this formalism is that when there's a nice geometric way to pass Lagrangians through a correspondence (for instance taking the preimage of $L\subset \mu^{-1}(0)/G$ in $\mu^{-1}(0)\subset M$) the geometric and formal approaches give quasi-isomorphic objects in $F^\sharp(Y)$. (Usually you can't pass a Lagrangian submanifold through a correspondence without making a terrible mess - hence the formal approach.)

For me, all this is exciting because we can at last compute Floer cohomology using its naturality properties, rather than by direct attack on the equations.

[In the new paper that inspired your question, the authors suggest that their F-M kernels should be coisotropic branes in the sense of Kapustin-Orlov. Floer cohomology for such branes is supposed to be some weird mixture of pseudo-holomorphic discs and Dolbeault cohomology over a holomorphic foliation - but there is no concrete proposal on the table
and for the moment this is just an intriguing idea. For the purposes of homological mirror symmetry, idempotent endomorphisms in the Fukaya category apparently provide the enlargement that K-O observed was necessary.]