Coupled models of geodynamics and landform evolution are receiving growing interest. While landform evolution models typically describe the response of the topography to vertical uplift and subsidence, horizontal movement becomes an essential component in combination with geodynamics. This study compares Eulerian and Lagrangian schemes for including horizontal movement and deformation of the crust in fluvial landform evolution models. As a main result, Eulerian schemes do not allow rivers to move perpendicularly to their main flow direction. In turn, they generate a strong artificial increase in surface elevation at high velocities, which makes them unsuitable for scenarios with strong horizontal movement. The Lagrangian approach avoids these problems. In turn, it is technically more complicated if deformation is so strong that remeshing is necessary. Remeshing is challenging for the widely used D8 topology, which derives the flow pattern from the 8 nearest and diagonal neighbors. Furthermore, remeshing causes artifacts by damming rivers temporarily. While these artifacts can be reduced technically, the question remains under which conditions remeshing is necessary. Our results suggest that the simple D8 topology can be used without remeshing for a shear strain of up to about 2, which is already a quite strong deformation.