Representing atmospheric transport of constituents accurately in a chemistry climate model is a challenge. This is true in particular for a realistic representation of atmospheric transport barriers, such as those at the edge of the polar vortices or at the tropopause. When transport is represented employing Lagrangian methods, numerical problems representing transport barriers may be obviated. A first implementation of a Lagrangian transport model (CLaMS) driven by horizontal winds and vertical velocities of the icosahedral nonhydrostatic model (ICON) using the Modular Earth Submodel System (MESSy) is presented in this study. The diabatic heating rates deduced from the temperature tendencies in the (free-running) ICON model allow vertical velocities to be determined and transport calculations in isentropic (diabatic) coordinates. The deduced diabatic heating rates agree qualitatively well with ERA5 reanalysis values in the zonal annual mean, but some discrepancies remain. The chemical transport is analyzed by zonal mean climatologies of nitrous oxide and compared with climatologies obtained from MLS observations. There is an overall agreement between the simulation and N2O observations by the Microwave Limb Sounder (MLS) satellite instrument. This is especially true for the N2O gradients at the edge of the polar vortex. The representation of the Antarctic vortex in the model is analyzed by calculations of horizontal gradients in the distribution of nitrous oxide. Overall, the Antarctic vortex and the associated transport barrier at its edge are well represented in the simulation, although the simulated polar vortex is larger than observed. Some differences between the observations and the Lagrangian simulation may be caused by the underlying ICON winds. The coupled ICON/MESSy-CLaMS transport scheme allows tracer distributions in the free troposphere and in the stratosphere to be better simulated than by classical Eulerian schemes.