We consider the Foldy-Whouthuysen (FW) transformation of the Dirac equation coupled to a background soliton field which is equivalent to a position-dependent mass m(x) such that at each limit x → ±∞, the mass to the left and to the right tends to a (possibly different) constant, with a sign difference at each side. We then build-up a third order unitarily transformed Schrödinger-like Hamiltonian as a counterpart of the corresponding to the well known Jackiw-Rebbi model. By further FW-transforming the Dirac spinor, we establish the relation between the non-relativistic and relativistic wave functions up to this order of approximation for generic position dependent mass profiles. For the economic choice m(x) = m0x/|x|, we find that these spinors are the same up to an overall constant.