All wave equations on a curved background can be solved in the eikonal approximation if the typical curvature length of the space-time is much larger than the wave-length of motion of the particle. If the equations are minimally coupled to the background, i.e. only through the d'Alembertian, the eikonal equation is to leading order equivalent to the equation of a geodesic of a corresponding particle. Obviously, if the typical curvature length is comparable to the wavelength, interference comes to play and the solutions may strongly disagree with a "geodesic streamlining" of the wave-packet.
But for example spinning matter is never minimally coupled to the background. A spinning classical particle actually interacts with the curved background in a non-trivial way and the interaction can be obtained by taking a zero-volume limit of a classical extended rigid body to yield the so-called Mathisson Papatreou equations
ddτ(muμ−uσ˙Sμσ)=−12RμνσρuνSρσ
˙Sμν=2u[μ˙Sν]ρuρ
where
Sμν represents the internal angular momentum of the particle (there are some supplementary technicalities which need not be discussed here). I.e. on a "pole-dipole" classical particle level, there is additional spin-curvature interaction. This is also true for a spinor field on a curved background where the square of the Dirac operator on a curved background yields
(◻+m2+R/4)ψ=0
That is, on any level there is non-minimal interaction between the curvature and spin. I have never investigated the connection between these two; feel free to dig, the keyword would probably be Frenkel and his 1926 theory of the spinning electron.
On the other hand, for example for a massless scalar it is possible to have a non-minimal coupling to curvature (◻+ξR)ϕ=0
where
ξ can be a non-zero constant. A rather popular choice is the conformal coupling
ξ=1/6 which under a conformal rescaling
g→˜g=Ωg allows to obtain solutions of the new equation from the original ones as
˜ϕ=Ω−1ϕ. To state the point of this paragraph differently, the translational invariance of Minkowski allowed us to determine the respective classical equations unambiguously but this is no longer true on a curved background, here the field equation may sometimes not reflect the classical geodesic -- even if no spin is involved.