  We study certain exactly marginal gaugings involving arbitrary numbers of
Argyres-Douglas (AD) theories and show that the resulting Schur indices are
related to those of certain Lagrangian theories of class $\mathcal{S}$ via
simple transformations. By writing these quantities in the language of 2D
topological quantum field theory (TQFT), we easily read off the $S$-duality
action on the flavor symmetries of the AD quivers and also find expressions for
the Schur indices of various classes of exotic AD theories appearing in
different decoupling limits. The TQFT expressions for these latter theories are
related by simple transformations to the corresponding quantities for certain
well-known isolated theories with regular punctures (e.g., the
Minahan-Nemeschansky $E_6$ theory and various generalizations). We then
reinterpret the TQFT expressions for the indices of our AD theories in terms of
the topology of the corresponding 3D mirror quivers, and we show that our
isolated AD theories generically admit renormalization group (RG) flows to
interacting superconformal field theories (SCFTs) with thirty-two (Poincar\'e
plus special) supercharges. Motivated by these examples, we argue that, in a
sense we make precise, the existence of RG flows to interacting SCFTs with
thirty-two supercharges is generic in a far larger class of 4D $\mathcal{N}=2$
SCFTs arising from compactifications of the 6D $(2,0)$ theory on surfaces with
irregular singularities.
