  A broad class of modified gravities can result in a modified friction effect
in the propagation of gravitational waves (GWs). This effect changes the
amplitude-damping rate of GWs during their propagation in the cosmological
distance and thus modifies the standard luminosity distance of GWs in general
relativity. Therefore, one can constrain this modified friction by measuring
both the luminosity distance and redshift of the GW sources. In this paper, we
investigate the prospects of constraining such modified friction effect by
using the precessing binary black holes with ground-based GW detectors. For
this purpose, we consider 20 precessing events detected by the GW detector
network consisting of two LIGO detectors and two third-generation GW detectors
(the Einstein Telescope and the Cosmic Explorer). The redshift information of
these events is obtained by identifying their possible host galaxies in the
GLADE+ galaxy catalog. We show that the precession in the binary system can
improve significantly the precision of luminosity distance and thus lead to a
tighter constraint on the modified friction. By assuming narrow priors on
cosmological parameters that are consistent with the uncertainties of Planck
2018 results, our analysis shows that the modified friction effect,
characterized by two parameters $(\Xi_0, n)$, can be constrained to be $\Xi_0 =
1.002^{+0.004}_{-0.004}$ and $n=3.257^{+2.595}_{-2.192}$, in which the result
of $\Xi_0$ is about two orders of magnitude better than current result from an
analysis with GWTC-3. Our result sets the stage for future research with
third-generation GW detectors, offering new insights into gravitational
parameter modifications. It also contributes to the understanding of the
properties and applications of binary black hole systems with precession.
