  Extreme mass-ratio inspirals (EMRIs), namely binary systems composed of a
massive black hole and a compact stellar-mass object, are anticipated to be
among the gravitational wave (GW) sources detected by the Laser Interferometer
Space Antenna (LISA). Similarly to compact binary mergers detected by current
GW detectors, EMRIs can be used as cosmic rulers to probe the expansion of the
Universe. Motivated by tensions in current cosmological observations as well as
by alternative models of dark energy, modified gravity theories can affect the
propagation of GWs across cosmological distances, with modifications commonly
parametrised in terms of two phenomenological parameters, $\Xi_0$ and $n$. In
this work we adopt a Bayesian approach to constrain for the first time
parametrised deviations from General Relativity using the loudest simulated
EMRIs detected by LISA as dark sirens with a simulated galaxy catalog. Assuming
all the cosmological parameters except $\Xi_0$ are already tightly constrained,
our forecasts show that $\Xi_0$ can be constrained to a few percent level (90%
C.I.) with 4 years of LISA observations, unless EMRI detection rates turn out
to be closer to current pessimistic expectations. These results quickly degrade
if additional cosmological parameters are inferred simultaneously, but become
more robust with an extended LISA observation period of 10 years. Overall, we
find that EMRIs with LISA are better at constraining modified GW propagation
than current second-generation ground-based GW detectors, but they will only be
comparable to third-generation detectors in the most optimistic scenarios.
