Atmospheric models used for practical climate simulation must be capable handling the transport of hundreds of tracers. For computational efficiency conservative multi-tracer semi-Lagrangian type transport schemes are appropriate. Global models based on high-order Galerkin approach employ highly non-uniform spectral-element grids, and semi-Lagrangian transport is a challenge on those grids. A conservative semi-Lagrangian scheme (SPELT - SPectral-Element Lagrangian Transport) employing a multi-moment compact reconstruction procedure is developed for non-uniform quadrilateral grids. The scheme is based on a characteristic semi-Lagrangian method that avoids complex and expensive upstream area computations. The SPELT scheme has been implemented in the High-Order Method Modeling Environment (HOMME), which is based on a cubed-sphere grid with spectral-element spatial discretization. Additionally, we show the (strong) scalability and multi-tracer efficiency using several benchmark tests. The SPELT solution can be made monotonic (positivity preserving) by combining the flux-corrected transport algorithm, which is demonstrated on a uniform resolution grid. In particular, SPELT can be efficiently used for non-uniform grids and provides accurate and stable results for high-resolution meshes.