Removing the continuity assumption, perhaps even time, kind resolves the duplicate question: both copies (if conscious) should have a memory of conscious you and experience their consciousness being a continuation of that, while you, now you, are as removed from those future copies, as you are from me.
One question, that seems tied to consciousness, is what is "now"? Are there objective reasons to believe in the existence of a canonical "now" existing independent of conscious experience? If not then there really is no point in distinguishing either momental or continuous experiences as special in any way. Perhaps consciousness just emerges as an interpretation of a single state connecting past and future by memory and anticipation, it's simply appears continuous like the series of images at the top of the article appears to be.