All I can think of is, on Sonic 3D Blast, Sonic must deliver flickies to special rings where they vanish. I'm guessing Flickies may have the ability to travel through portals.
Couldn't anyone teleport with that special rings, not just Flickies?
At least in Sonic movies they use that kind of scene a lot.

There could be various ways for Enigma to copy their moves. I was surprised when IDW Sonic comics introduced Power Cores as plot-relevant. It also shows that the use of Power Cores are not strictly to an individuals' "class type". The blue Power Core seem to offer a burst of speed to the user and the yellow Power Core offers a boost in acrobatic skill even if the user can't fly.
So it could be possible Enigma is using Power Cores to replicate skills of the characters she's disguised as.
Yeah, didn't expect Power Cores would be used again.
Only showed up in two games, right? (Sonic Heroes & London 2012 Olympic Games)