The motion and deformation of a spherical elastic capsule freely flowing in a pore of comparable dimension is studied. The thin capsule membrane has a neo-Hookean shear softening constitutive law. The three-dimensional fluid–structure interactions are modelled by coupling a boundary integral method (for the internal and external fluid motion) with a finite element method (for the membrane deformation). In a cylindrical tube with a circular cross-section, the confinement effect of the channel walls leads to compression of the capsule in the hoop direction. The membrane then tends to buckle and to fold as observed experimentally. The capsule deformation is three-dimensional but can be fairly well approximated by an axisymmetric model that ignores the folds. In a microfluidic pore with a square cross-section, the capsule deformation is fully three-dimensional. For the same size ratio and flow rate, a capsule is more deformed in a circular than in a square cross-section pore. We provide new graphs of the deformation parameters and capsule velocity as a function of flow strength and size ratio in a square section pore. We show how these graphs can be used to analyse experimental data on the deformation of artificial capsules in such channels.