  Barlowite Cu$_4$(OH)$_6$FBr shows three-dimensional (3D) long-range
antiferromagnetism, which is fully suppressed in Cu$_3$Zn(OH)$_6$FBr with a
kagome quantum spin liquid ground state. Here we report systematic studies on
the evolution of magnetism in the Cu$_{4-x}$Zn$_x$(OH)$_{6}$FBr system as a
function of $x$ to bridge the two limits of Cu$_4$(OH)$_6$FBr ($x$=0) and
Cu$_3$Zn(OH)$_6$FBr ($x$=1). Neutron-diffraction measurements reveal a
hexagonal-to-orthorhombic structural change with decreasing temperature in the
$x$ = 0 sample. While confirming the 3D antiferromagnetic nature of
low-temperature magnetism, the magnetic moments on some Cu$^{2+}$ sites on the
kagome planes are found to be vanishingly small, suggesting strong frustration
already exists in barlowite. Substitution of interlayer Cu$^{2+}$ with
Zn$^{2+}$ with gradually increasing $x$ completely suppresses the bulk magnetic
order at around $x$ = 0.4, but leaves a local secondary magnetic order up to
$x\sim 0.8$ with a slight decrease in its transition temperature. The
high-temperature magnetic susceptibility and specific heat measurements further
suggest that the intrinsic magnetic properties of kagome spin liquid planes may
already appear from $x>0.3$ samples. Our results reveal that the
Cu$_{4-x}$Zn$_x$(OH)$_6$FBr may be the long-thought experimental playground for
the systematic investigations of the quantum phase transition from a long-range
antiferromagnet to a topologically ordered quantum spin liquid.
