We consider a general setting for dynamic tensor field tomography in an inhomogeneous refracting and absorbing medium as inverse source problem for the associated transport equation. Following Fermat's principle the Riemannian metric in the considered domain is generated by the refractive index of the medium. There is wealth of results for the inverse problem of recovering a tensor field from its longitudinal ray transform in a static euclidean setting, whereas there are only few inversion formulas and algorithms existing for general Riemannian metrics and time-dependent tensor fields. It is a well-known fact that tensor field tomography is equivalent to an inverse source problem for a transport equation where the ray transform serves as given boundary data. We prove that this result extends to the dynamic case. Interpreting dynamic tensor tomography as inverse source problem represents a holistic approach in this field. To guarantee that the forward mappings are well-defined, it is necessary to prove existence and uniqueness for the underlying transport equations. Unfortunately, the bilinear forms of the associated weak formulations do not satisfy the coercivity condition. To this end we transfer to viscosity solutions and prove their unique existence in appropriate Sobolev (static case) and Sobolev-Bochner (dynamic case) spaces under a certain assumption that allows only small variations of the refractive index. Numerical evidence is given that the viscosity solution solves the original transport equation if the viscosity term turns to zero.Keywords attenuated refractive dynamic ray transform of tensor fields • geodesics • transport equation • viscosity solutions TFT has many possible applications. One is the reconstruction of velocity fields of liquids and gases. This can be used, e.g., to represent blood flows in medicine. TFT is also used in electron tomography, industry, geo-and astrophysics to name only a few application fields. Pioneered by Norton [20] in 1988, fundamental results on Doppler tomography