My guess is that many mainland species are in a dynamic equilibrium with other species along some dimension X, such that too little or too much X is a disadvantage, and the species stabilizes at some X=Xo.

When such a species accidentally ends up in a much smaller ecosystem, the probability of finding a substitute stabilizing ecological force is lower than if it were to end up on a different continent of equal size. So there's more chance of runaway evolution along dimension X. So relative to Xo, the new equilibrium, Xo' is abnormal. More specifically, it might end up at a physics rather than biology extremum, at a boundary of the design space expressible within its genome, limited by something like weight to volume ratio or thermodynamic efficiency of feeding, rather than an "interior point."

A related phenomenon is how whales grew huge. The land mammals that they descended from were likely stabilized at smaller sizes by forces that don't exist in water.

I am pretty sure I could whip up a simulation where this happens, but that would not be a real proof. You'd need empirical data on 'substitution probability' numbers between island and continental species. For example, you'd ask if 'cat' maps to an equivalent.