  Modifications of General Relativity leave their imprint both on the cosmic
expansion history through a non-trivial dark energy equation of state, and on
the evolution of cosmological perturbations in the scalar and in the tensor
sectors. In particular, the modification in the tensor sector gives rise to a
notion of gravitational-wave (GW) luminosity distance, different from the
standard electromagnetic luminosity distance, that can be studied with standard
sirens at GW detectors such as LISA or third-generation ground based
experiments. We discuss the predictions for modified GW propagation from some
of the best studied theories of modified gravity, such as Horndeski or the more
general degenerate higher order scalar-tensor (DHOST) theories, non-local
infrared modifications of gravity, bigravity theories and the corresponding
phenomenon of GW oscillation, as well as theories with extra or varying
dimensions. We show that modified GW propagation is a completely generic
phenomenon in modified gravity. We then use a simple parametrization of the
effect in terms of two parameters $(\Xi_0,n)$, that is shown to fit well the
results from a large class of models, to study the prospects of observing
modified GW propagation using supermassive black hole binaries as standard
sirens with LISA. We construct mock source catalogs and perform detailed Markov
Chain Monte Carlo studies of the likelihood obtained from LISA standard sirens
alone, as well as by combining them with CMB, BAO and SNe data to reduce the
degeneracies between cosmological parameters. We find that the combination of
LISA with the other cosmological datasets allows one to measure the parameter
$\Xi_0$ that characterizes modified GW propagation to the percent level
accuracy, sufficient to test several modified gravity theories. [Abridged]
