To a distant observer a gravitating body increases in mass only by the energy of particles which fall into it as measured at the defined distant zero position. Gravitational potential energy converts to kinetic energy, or blueshift in frequency, as a particle approaches the gravitating mass and returns to latency, or redshifts in frequency, if the particle climbs back out of the gravity well. Consider that a nucleus on the surface of a gravitating mass holds a fixed position relative to that body. Similarly, over a short distance traveled, a nucleus in an accretion disk around a spinning black hole can be considered to be undergoing centrifugal acceleration about equal to and just counter to its free fall geodesic path toward the singularity. 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.