With an immersed-boundary lattice-Boltzmann method, we consider the transit of a three-dimensional initially spherical capsule with a viscoelastic membrane through a cross-slot microchannel. The capsule is released with a small initial off-centre distance in the feeding channel, to mimic experiments where capsules or cells are not perfectly aligned with the centreline. Our main objective is to establish the phase diagram of the capsule's deformation modes as a function of the flow inertia and capsule membrane viscosity. We mainly find three deformation modes in the channel cross-slot. For a capsule with low membrane viscosity, a quasi-steady mode occurs at low Reynolds numbers ( increasing to 20, an overshoot–retract mode is observed. The capsule deformation oscillates on an inertial–elastic time scale, suggesting that the dynamics is mainly driven by the balance of the inertial and membrane elastic forces. The membrane viscosity slows down the capsule deformation and suppresses the overshoot–retract mode. A capsule with high membrane viscosity undergoes a continuous-elongation mode, in which its deformation keeps increasing during most of its journey in the channel cross-slot. We summarise the results in phase diagrams, and propose a scaling model which can predict the deformation modes of a viscoelastic capsule in the inertial flow regime. We also discuss implications of the present findings for practical experiments for mechanical characterisation of capsules or cells.