The authors define the conditional generalized Fourier-Feynman transform and conditional generalized convolution product for functionals on a function space which is rather general than the Wiener space and was introduced by \textit{S. J. Chang} and \textit{D. Skoug} [Integral Transforms Spec. Funct. 14, No. 5, 375--393 (2003; Zbl 1043.28014)]. Then they establish some relationships between the two concepts for functionals which belong to a Banach algebra in the function space. \textit{S. J. Chang} and \textit{D. Skoug} [loc. cit.] used a generalized Brownian motion process to define a generalized analytic Feynman integral and a generalized analytic Fourier-Feynman transform. The Wiener process used by \textit{C. Park} and \textit{D. Skoug} [J. Korean Math. Soc. 38, No. 1, 61--76 (2001; Zbl 1015.28016)] is free of drift and is stationary in time while the stochastic process used in this paper is nonstationary in time and is subject to a drift.