We develop neutral-fermionic constructions for the factorial $gp$-and $gq$-functions introduced by Nakagawa and Naruse, which are respectively dual to the factorial $GQ$- and $GP$-functions of Ikeda and Naruse. In particular, we realize the factorial $GP$-, $GQ$- and $gq$-functions as vacuum expectation values. As applications, we obtain, Jacobi--Trudi type determinantal formulas for the transition coefficients between functions with different equivariant parameters for $gq$ and its dual $GP$, as well as a Pfaffian formula for the factorial $gq$-functions. We further prove a remarkable coincidence among the transition coefficients for parameter changes for $gp$, $gq$, $GQ$, and $GP$. These coefficients admit a description in terms of factorial Grothendieck polynomials of type A.