  The geodesic ray transform, the mixed ray transform and the transverse ray
transform of a tensor field on a manifold can all be seen as what we call
mixing ray transforms, compositions of the geodesic ray transform and an
invertible linear map on tensor fields. We show that the characterization of
the kernel and the stability of a mixing ray transform can be reduced to the
same properties of any other mixing ray transform. Our approach applies to
various geometries and ray transforms, including the light ray transform. In
particular, we extend studies in de Hoop--Saksala--Zhai (2019) from compact
simple surfaces to orientable surfaces with solenoidally injective geodesic ray
transform. Our proofs are based on algebraic arguments.
