  Recently an algorithm to dualize a theory into its mirror dual has been
proposed, both for $3d$ $\mathcal{N}=4$ linear quivers and for their $4d$
$\mathcal{N}=1$ uplift. This mimics the manipulations done at the level of the
Type IIB brane setup that engineers the $3d$ theories, where mirror symmetry is
realized as $S$-duality, but it is enirely field-theoretic and based on the
application of genuine infra-red dualities that implement the local action of
$S$-duality on the quiver. In this paper, we generalize the algorithm to the
full duality group, which is $SL(2,\mathbb{Z})$ in $3d$ and $PSL(2,\mathbb{Z})$
in $4d$. This also produces dualities for $3d$ $\mathcal{N}=3$ theories with
Chern--Simons couplings, some of which have enhanced $\mathcal{N}=4$
supersymmetry, and their new $4d$ $\mathcal{N}=1$ counterpart. In addition, we
propose three ways to study the RG flows triggered by possible VEVs appearing
at the last step of the algorithm, one of which uses a new duality that
implements the Hanany--Witten move in field theory.
