  The quantum spin systems Cu$_2$M'BO$_5$ (M' = Al, Ga) with the ludwigite
crystal structure consist of a structurally ordered Cu$^{2+}$ sublattice in the
form of three-leg ladders, interpenetrated by a structurally disordered
sublattice with a statistically random site occupation by magnetic Cu$^{2+}$
and nonmagnetic Ga$^{3+}$ or Al$^{3+}$ ions. A microscopic analysis based on
density-functional-theory calculations for Cu$_2$GaBO$_5$ reveals a frustrated
quasi-two-dimensional spin model featuring five inequivalent antiferromagnetic
exchanges. A broad low-temperature $^{11}$B nuclear magnetic resonance points
to a considerable spin disorder in the system. In zero magnetic field,
antiferromagnetic order sets in below $T_\text{N}$ $\approx$ 4.1 K and ~2.4 K
for the Ga and Al compounds, respectively. From neutron diffraction, we find
that the magnetic propagation vector in Cu$_2$GaBO$_5$ is commensurate and lies
on the Brillouin-zone boundary in the (H0L) plane, $\mathbf{q}_\text{m}$ =
(0.45 0 -0.7), corresponding to a complex noncollinear long-range ordered
structure with a large magnetic unit cell. Muon spin relaxation is monotonic,
consisting of a fast static component typical for complex noncollinear spin
systems and a slow dynamic component originating from the relaxation on
low-energy spin fluctuations. Gapless spin dynamics in the form of a diffuse
quasielastic peak is also evidenced by inelastic neutron scattering. Most
remarkably, application of a magnetic field above 1 T destroys the static
long-range order, which is manifested in the gradual broadening of the magnetic
Bragg peaks. We argue that such a crossover from a magnetically long-range
ordered state to a spin-glass regime may result from orphan spins on the
structurally disordered magnetic sublattice, which are polarized in magnetic
field and thus act as a tuning knob for field-controlled magnetic disorder.
