We establish a generic, fully-relativistic formalism to study
gravitational-wave emission by extreme-mass-ratio systems in
spherically-symmetric, non-vacuum black-hole spacetimes. The potential
applications to astrophysical setups range from black holes accreting baryonic
matter to those within axionic clouds and dark matter environments, allowing to
assess the impact of the galactic potential, of accretion, gravitational drag
and halo feedback on the generation and propagation of gravitational-waves. We
apply our methods to a black hole within a halo of matter. We find fluid modes
imparted to the gravitational-wave signal (a clear evidence of the black hole
fundamental mode instability) and the tantalizing possibility to infer galactic
properties from gravitational-wave measurements by sensitive, low-frequency
detectors.
