Degree divisibility in character correspondences

Recent results of the author prove the existence of bijections of certain sets of irreducible characters which satisfy many nice properties. In particular, these correspondences imply a strengthened version of the McKay Conjecture for solvable finite groups which takes into account fields of values and Schur indices. In the present paper, we prove that these correspondences have also the additional property that the degree of the corresponding character divides the degree of the original character, and we say something about the quotient as well.