I think @Gilphonmade some really good points on why this has to be using Transportation somehow.
The red-violet light that chases Kaladin reminded me of one of Dalinar's visions, the one where he is running around in the purelake and spots a spren with red eyes in the water. The Radiant there says something about the spren needing a carrier or an escort. Then a bigger spren shows up and turns into a thunderclast.
When Dalinar first sees the smaller spren he says it looks like a red spot in the water then it swims away.
So the bigger spren is one of the fused right? They are what animate stone to make a thunderclast?
The connection the Radiant mentions got me thinking what if the fused who use surges can make similar connections with spren.
Maybe the new fused Kaladin fought is using Transportation to leave the physical realm and then his little spren is his anchor. So the spren chases Kaladin around and lets the fused know when to come back. Maybe the spren is carrying the gem heart while he moves? And that's why it can only move so fast. Otherwise I think it would be able to fly around as fast as Syl.
I think this would be cool way for Transportation to work but I haven't thought of any good reasons for why the fused has to regrow a body. When Jasnah used Transportation it was very different.