On partition functions of SCFTs of class $\mathcal{S}$ (Gaiotto theories)

Question

For which theories of class $\mathcal{S}$ can we write down partition functions (as we can for Lagrangian theories), either in 4d or in the dual 2d CFT description? 

What is known about the $SL(2,\mathbb{Z})$ properties of the partition functions and what is the current progress on this direction?

Any specific references are welcome.

Comments

Let me comment that there has been a lot of work for the supersymmetric indices of such theories on $\mathbb{R}^3 \times \mathbb{S}^1$ and $\mathbb{S}^3  \times \mathbb{S}^1$. What is the difficulty though for the 4d partition functions?