  The change in mass of a nucleon, arising from its interactions with other
nucleons inside the target, results in velocity-dependent terms in the
Schr\"odinger equation that describes nucleon scattering. It has recently been
suggested in a number of publications that introducing and fitting
velocity-dependent terms improves the quality of the description of nucleon
scattering data for various nuclei. The present paper discusses
velocity-dependent optical potentials in a context of a three-body problem used
to account for deuteron breakup in the entrance channel of $(d,p)$ reactions.
Such potentials form a particular class of nonlocal optical potentials which
are a popular object of modern studies. It is shown here that because of a
particular structure of the velocity-dependent terms the three-body problem can
be formulated in two different ways. Solving this problem within an adiabatic
approximation results in a significant difference between the two approaches
caused by contributions from the high $n$-$p$ momenta in deuteron in one of
them. Solving the three-body problem beyond the adiabatic approximation may
remove such contributions, which is indirectly confirmed by replacing the
adiabatic approximation by the folding Watanabe model where such contributions
are suppressed. Discussion of numerical results is carried out for the
$^{40}$Ca($d,p)^{41}$Ca reaction where experimental data both on elastic
scattering in entrance and exit channels and on nucleon transfer are available.
