Does any part of our conscious awareness remain consistent through Time?

The other day I was watching a video by Susan Blackmore. She believes that our consciousness/"self" is an illusion a label. That essentially a consciousness is constantly coming into being and then dying every time our thoughts change.

This is a disturbing concept to me. The idea that you exist but in a second you'll pop out of existence and be replaced by another consciousness. Also it doesn't feel like that consciousness seems to be continuous and maybe there are slight changes but I don't know if I would call a completely different consciousness. I think I would say awareness/consciousness remains the same but new thoughts are always popping into it.

My question is is there anything about consciousness that remains unchanging throughout the time, or for long periods of time?