Frustrated magnetism on the kagome lattice has been a fertile ground for rich
and fascinating physics, ranging from experimental evidence of a spin liquid
to theoretical predictions of exotic superconductivity. Among experimentally
realized spin-12 kagome magnets, herbertsmithite, kapellasite, and haydeeite
[(Zn,Mg)Cu3(OH)6Cl2] are all well described by a three-parameter Heisenberg
model, but they exhibit distinctly different physics. We address the problem
using a pseudofermion functional renormalization-group approach and analyze
the low-energy physics in the experimentally accessible parameter range. Our
analysis places kapellasite and haydeeite near the boundaries between
magnetically ordered and disordered phases, implying that slight modifications
could dramatically affect their magnetic properties. Inspired by this, we
perform \textit{ab initio} density functional theory calculations of
(Zn,Mg,Cd)Cu3 (OH)6Cl2 at various pressures. Our results suggest that by
varying pressure and composition one can traverse a paramagnetic regime
between different magnetically ordered phases