A detailed calculation of frequency- and wave-vector-dependent correlation functions for an arbitrary tracer diffusing in a regular crystal against a background of hopping classical particles has recently been given by Tahir-Kheli and Elliott [Phys. Rev. B 27, 844 (1983)]. Here we present an important generalization of this work to a system with a dynamic background consisting of two arbitrary species of particles. In particular, the generalization includes a system where the tracer concentration itself is finite while an arbitrary concentration of other atoms is also present in the dynamic stream. The theory is exact to the leading nontrivial order in particle concentration ${x}_{A}$ and ${x}_{B}$. In the intermediate-concentration regime, the theory incorporates dominant fluctuations from the mean field. The present model can serve to usefully describe incoherent neutron scattering in metal-hydride interstitial solutions such as $M{A}_{{x}_{A}}{B}_{{x}_{B}}$ with $A,B\ensuremath{\equiv}\mathrm{H},\phantom{\rule{0ex}{0ex}}\mathrm{D},\phantom{\rule{0ex}{0ex}}\mathrm{a}\mathrm{n}\mathrm{d}\phantom{\rule{0ex}{0ex}}\mathrm{T}$ and $M\ensuremath{\equiv}\mathrm{Pd}\mathrm{and}\mathrm{Ti}$. Moreover, it can be used to treat tracer diffusion dynamics in nonstoichiometric metal oxides and, somewhat more simplistically, ionic conduction in the superionic state.