Particle suspensions are present in a wide variety of practical settings. Modelling these numerically is a challenging task that often requires the combination of multiple methodologies. This paper examines particle transport within a temperature-dependent viscosity fluid utilising a coupled approach of the lattice Boltzmann method and the discrete element method. This technique takes advantage of the locality of the lattice Boltzmann method to allow both the individual particle behaviour to be fully resolved and to permit fine-scale variation of fluid viscosity throughout the tested domains. It is firstly shown that a total energy conserving form of the lattice Boltzmann method is needed to accurately reconstruct the non-linear temperature profiles observed on Couette flows of fluids with changing viscosity. This model is then coupled to the discrete element method to demonstrate the quantitative and qualitative changes to particle motion that arise in channel-based geometries in the presence of a temperature-dependent viscosity fluid exposed to a constant temperature gradient. In particular, it is demonstrated that the particles settled faster in such and appear less likely to deviate into side channels in the presence of such fluids. These results demonstrate that temperature-dependent viscosity requires special consideration to be simulated correctly and does have quantitative impact on particle transport. This impact should be considered in models of fluids of changing temperature.