  We study the impact of gas accretion on the orbital evolution of black-hole
binaries initially at large separation in the band of the planned Laser
Interferometer Space Antenna (LISA). We focus on two sources:
(i)~stellar-origin black-hole binaries~(SOBHBs) that can migrate from the LISA
band to the band of ground-based gravitational-wave observatories within
weeks/months; and (ii) intermediate-mass black-hole binaries~(IMBHBs) in the
LISA band only. Because of the large number of observable gravitational-wave
cycles, the phase evolution of these systems needs to be modeled to great
accuracy to avoid biasing the estimation of the source parameters. Accretion
affects the gravitational-wave phase at negative ($-4$) post-Newtonian order,
and is therefore dominant for binaries at large separations. If accretion takes
place at the Eddington or at super-Eddington rate, it will leave a detectable
imprint on the dynamics of SOBHBs. In optimistic astrophysical scenarios, a
multiwavelength strategy with LISA and a ground-based interferometer can detect
about $10$ (a few) SOBHB events for which the accretion rate can be measured at
$50\%$ ($10\%$) level. In all cases the sky position can be identified within
much less than $0.4\,{\rm deg}^2$ uncertainty. Likewise, accretion at $\gtrsim
10\%$ ($\gtrsim 100\%$) of the Eddington rate can be measured in IMBHBs up to
redshift $z\approx 0.1$ ($z\approx 0.5$), and the position of these sources can
be identified within less than $0.01\,{\rm deg}^2$ uncertainty. Altogether, a
detection of SOBHBs or IMBHBs would allow for targeted searches of
electromagnetic counterparts to black-hole mergers in gas-rich environments
with future X-ray detectors (such as Athena) and radio observatories (such as
SKA).
