Is there continuity of consciousness during transport?
Yes. We've *seen* it. In "Realm of Fear" we see the entire transport process, from Barclay's POV. It's an unbroken 'stream' of consciousness. Barclay is awake and aware the whole time.
As for being dematerialized and all that: In a way, our bodies go through that every second of every day. It's called aging.