Summary: We propose an extension of semi-implicit models for the shallow water or Euler equations on staggered Cartesian meshes to handle the presence of cut cells at the boundaries of the computational domain. Semi-Lagrangian advection algorithms are also extended to staggered Cartesian meshes with cut boundary cells. A linear reconstruction algorithm and radial basis function interpolation are used, respectively, to compute approximate trajectories and to recover an accurate velocity field in cells intersected by the boundary. The effective accuracy of radial basis function interpolators for semi-Lagrangian algorithms is assessed by simple numerical experiments. Timestep control techniques in the computation of the semi-Lagrangian trajectories are also discussed. The accuracy of the resulting models for environmental flows is demonstrated by a number of test cases simulating nonlinear open channel flows and baroclinic flows over an isolated obstacle.