There is no prove that continuity really matters. In fact who goes to say that your current conscious self is the same as your self from five minutes ago. After all you do not feel what they felt. The only thing that makes you think that the two are related or even the same is your state (memories, emotions). Why would we even think about whether cloning is able to transmit consciousness when we don’t even know if consciousness is transmitted over time?

Edit: Just to clarify my opinion: This means that the relationship between my self from five seconds ago and my current self and the relationship between my self from five seconds ago and a clone of that self that aged the same amount would be equivalent. Both of us would _not_ be the same as my past self