  The detection of gravitational waves (GWs) propagating through cosmic
structures can provide invaluable information on the geometry and content of
our Universe, as well as on the fundamental theory of gravity. In order to test
possible departures from General Relativity, it is essential to analyse, in a
modified gravity setting, how GWs propagate through a perturbed cosmological
space-time. Working within the framework of geometrical optics, we develop
tools to address this topic for a broad class of scalar-tensor theories,
including scenarios with non-minimal, derivative couplings between scalar and
tensor modes. We determine the corresponding evolution equations for the GW
amplitude and polarization tensor. The former satisfies a generalised evolution
equation that includes possible effects due to a variation of the effective
Planck scale; the latter can fail to be parallely transported along GW
geodesics unless certain conditions are satisfied. We apply our general
formulas to specific scalar-tensor theories with unit tensor speed, and then
focus on GW propagation on a perturbed space-time. We determine corrections to
standard formulas for the GW luminosity distance and for the evolution of the
polarization tensor, which depend both on modified gravity and on the effects
of cosmological perturbations. Our results can constitute a starting point to
disentangle among degeneracies from different sectors that can influence GW
propagation through cosmological space-times.
