  We revisit the predictions for the merger rate of massive black hole binaries
detectable by the Laser Interferometer Space Antenna (LISA) and their
background signal for pulsar-timing arrays. We focus on the effect of the
delays between the merger of galaxies and the final coalescence of black hole
binaries, and on supernova feedback on the black hole growth. By utilizing a
semi-analytic galaxy formation model, not only do we account for the driving
the evolution of binaries at separations $\lesssim 1$ pc (gas-driven migration,
stellar hardening and triple/quadruple massive black hole systems), but we also
improve on previous studies by accounting for the time spent by black hole
pairs from kpc down to pc separation. We also include the effect of supernova
feedback, which may eject gas from the nuclear region of low-mass galaxies,
thus hampering the growth of black holes via accretion and suppressing their
orbital migration in circumbinary disks. Despite including these novel physical
effects, we predict that the LISA detection rate should still be $\gtrsim 2
\mbox{yr}^{-1}$, irrespective of the model for the black hole seeds at high
redshifts. Scenarios where black holes form from $\sim100 M_\odot$ seeds are
more significantly impacted by supernova feedback. We also find that for
detectable events, the merging black holes typically have mass ratios between
$\sim 0.1$ and $1$. Predictions for the stochastic background in the band of
pulsar-timing array experiments are instead rather robust, and show only a mild
dependence on the model.
