Shoaling solitary internal waves are ubiquitous features in the coastal regions of the world's oceans where waves with a core of recirculating fluid (trapped cores) can provide an effective transport mechanism. Here, numerical evidence is presented which suggests that there is a close connection between the limiting behaviour of large-amplitude solitary waves and the formation of such waves via shoaling. For some background states, large-amplitude waves are broad, having a nearly horizontal flow in their centre. The flow in the centre of such waves is called a conjugate flow. For other background states, large-amplitude waves can reach the breaking limit, at which the maximum current in the wave is equal to the wave's propagation speed. The presence of a background current with near-surface vorticity of the same sign as that induced by the wave can change the limiting behaviour from the conjugate-flow limit to the breaking limit. Numerical evidence is presented here which suggests that if large solitary waves cannot reach the breaking limit in the shallow water, that is if the background flow has a conjugate flow, then waves with trapped cores will not be formed via shoaling. It is also shown that, due to a change in the limiting behaviour of large waves, an appropriate background current can enable the formation of waves with trapped cores in stratifications for which such waves are not formed in the absence of a background current.