For all practical purposes each holds a fixed position nearby and relative to its associated gravitating mass. Neutrinos carry kinetic energy and nothing, or next to nothing else. Much of the kinetic energy carried by a neutrino converts to rest mass of the daughter particles if the neutrino interacts with a nucleus. For example, for Al27, neutrino in, beta negative out, Si27, the proton rich silicon atom is about 5.0 MeV more massive than the Al27 atom - mass derived from the kinetic energy of the neutrino. The weak nuclear force mediating this interaction cannot discriminate between the original kinetic energy and the blueshift portion of the neutrino's total kinetic energy in the frame of reference of the nucleus observer described above.
Kinetic energy is kinetic energy, period.
Therefore gravitational potential energy converts to kinetic energy as the freely falling neutrino approaches the gravitating body and then converts to rest mass if the neutrino interacts with a nucleus fixed in position relative to that gravitating body. Rest mass is observable in all reference frames and does contribute to the increase in mass of the gravitating body upon particle accretion. This is very simple. There are just two particles to consider and two different energy outcomes depending only on whether they interact near the gravitating body or not. There is more to this of course.