  We have systematically studied the magnetic properties of
Cu$_{4-x}$Zn$_x$(OH)$_6$FBr by the neutron diffraction and muon spin rotation
and relaxation ($\mu$SR) techniques. Neutron-diffraction measurements suggest
that the long-range magnetic order and the orthorhombic nuclear structure in
the $x$ = 0 sample can persist up to $x$ = 0.23 and 0.43, respectively. The
temperature dependence of the zero-field (ZF) $\mu$SR spectra provide two
characteristic temperatures, $T_{A0}$ and $T_{\lambda}$. Comparison between
$T_{A0}$ and $T_M$ from previously reported magnetic-susceptibility
measurements suggest that the former comes from the short-range interlayer-spin
clusters that persist up to $x$ = 0.82. On the other hand, the doping level
where $T_{\lambda}$ becomes zero is about 0.66, which is much higher than
threshold of the long-range order, i.e., $\sim$ 0.4. Our results suggest that
the change in the nuclear structure may alter the spin dynamics of the kagome
layers and a gapped quantum-spin-liquid state may exist above $x$ = 0.66 with
the perfect kagome planes.
