Fluid and ultralight bosonic dark matter can interact through gravity to form stable fermion-boson stars, which are static and regular mixed solutions of the Einstein-Euler-(complex, massive)-Klein-Gordon system. In this work we study the dynamical formation via gravitational cooling of a spherical mixed white-dwarf-boson star, whose properties depend on the boson particle mass and the mass of the boson star. Due to the accretion of bosonic dark matter, the white dwarf migrates to a denser and more compact object with a boson star core, thus modifying its gravitational redshift and altering the electromagnetic radiation emitted from the photosphere. We discuss the implications of the changes in the gravitational redshift that in principle could be produced by any type of dark matter and that might lead to small discrepancies in the estimation of masses and radii derived from white-dwarf observations.