<p>We study the non-geodesic motion of massive particles and the evolution of kinematical quantities within the framework of <i>f</i>(<i>G</i>) modified gravity non-minimally coupled to an imperfect fluid. An explicit coupling between the matter Lagrangian and an arbitrary function of the Gauss–Bonnet invariant <i>G</i> gives rise to an extra force that is orthogonal to the fluid four-velocity. Starting from a variational principle, we derive the modified field equations and obtain the explicit form of this force from the divergence of the energy–momentum tensor. We then establish a generalized Raychaudhuri equation that includes the effects of bulk and shear viscosities, heat flux, and the non-minimal coupling. This formulation extends previous analyses in <i>f</i>(<i>R</i>) gravity with perfect fluids and uncovers new dynamical features induced by geometry–matter interactions. The physical implications are discussed in relation to non-geodesic flows, cosmic evolution, and possible observational consequences in gravitational physics.</p>