The bulk motion of the gas would contribute to the relativistic mass-energy of the compact object.
White dwarfs are not produced by collapse, so there is nothing to comment on here.
Neutron stars are produced by collapse. Most of the kinetic energy is lost in the form of neutrinos, the rest would be thermalised within the neutron star, increasing the kinetic energy of its constituent particles (which become partially relativistic) and that energy does contribute to the gravitational mass.
When a black hole forms then all the mass-energy that goes into its formation contributes to its gravitational mass.
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