  We report a comprehensive muon spectroscopy study of the Zn-barlowite series
of $S={\frac{1}{2}}$ kagom\'e antiferromagnets, Zn$_x$Cu$_{4-x}$(OH)$_{6}$FBr,
for $x=0.00$ to $0.99(1)$. By combining muon spin relaxation and rotation
measurements with state-of-the-art density-functional theory muon-site
calculations, we observe the formation of both $\mu$--F and $\mu$--OH complexes
in Zn-barlowite. From these stopping sites, implanted muon spins reveal the
suppression of long-range magnetic order into a possible quantum spin liquid
state upon increasing concentration of Zn-substitution. In the parent compound
($x=0$), static long-range magnetic order below $T_{\mathsf{N}}=15$ K manifests
itself in the form of spontaneous oscillations in the time-dependent muon
asymmetry signal consistent with the dipolar fields expected from the
calculated muon stopping sites and the previously determined magnetic structure
of barlowite. Meanwhile, in the $x=1.0$ end-member of the series---in which
antiferromagnetic kagom\'e layers of Cu$^{2+}$ $S={\frac{1}{2}}$ moments are
decoupled by diamagnetic Zn$^{2+}$ ions---we observe that dynamic magnetic
moment fluctuations persist down to at least 50 mK, indicative of a quantum
disordered ground state. We demonstrate that this crossover from a static to
dynamic magnetic ground state occurs for compositions of Zn-barlowite with
$x>0.5$, which bears resemblance to dynamical behaviour of the widely studied
Zn-paratacamite series that contains the quantum spin liquid candidate
herbertsmithite.
