  Primordial black holes in the mass range of ground-based gravitational-wave
detectors can comprise a significant fraction of the dark matter. Mass and spin
measurements from coalescences can be used to distinguish between an
astrophysical or a primordial origin of the binary black holes. In standard
scenarios the spin of primordial black holes is very small at formation.
However, the mass and spin can evolve through the cosmic history due to
accretion. We show that the mass and spin of primordial black holes are
correlated in a redshift-dependent fashion, in particular primordial black
holes with masses below ${\cal O}(30)M_\odot$ are likely non-spinning at any
redshift, whereas heavier black holes can be nearly extremal up to redshift
$z\sim10$. The dependence of the mass and spin distributions on the redshift
can be probed with future detectors such as the Einstein Telescope. The mass
and spin evolution affect the gravitational waveform parameters, in particular
the distribution of the final mass and spin of the merger remnant, and that of
the effective spin of the binary. We argue that, compared to the
astrophysical-formation scenario, a primordial origin of black hole binaries
might better explain the spin distribution of merger events detected by
LIGO-Virgo, in which the effective spin parameter of the binary is compatible
to zero except possibly for few high-mass events. Upcoming results from
LIGO-Virgo third observation run might reinforce or weaken these predictions.
