What makes you think that you have an uninterrupted consciousness now?
I am pretty sure that humans have interruptions of brain activity all the time on such a small time scale that we don't notice any gaps...
Your consciousness doesn't extend into the future or into the past, allthough you can access memories from the past which are stored in your brain.
You always live in the present, which you can't even pinpoint as a specific point in time.
My future self does not exist, there are just possibilities and probabilities.
When I step into a transporter and get rematerialized on a starship in orbit I will be no different than before: a person with the same memories up to the transport the same arrangement of atoms and molecules.
Sure, I would be a different person from that point on compared to the possible person that didn't get beamed up that might have been, but that would be no different than the difference between a person going right at the next corner from him going left instead.
That is why the episode with the Riker clone is so great because they really are the same person and none of them has a better claim on "Riker's" life. It was a freakish accident that resulted in two possible existences of the same person.
In a way it is sort of a locally confined alternate timeline case where both Rikers are trapped in the same reality.
The whole thing only becomes problematic if you believe in a human soul, an independent undefined entity from your corporeal existance.
There are cases where it could be argued that the person stepping into the transporter has been killed because the emerging person is clearly not an exact duplicate.
The deaging of Pulaski; the aging shenanigans of Picard, Ro and Guinan; to a lesser degree the backup beaming of Picard when he was possessed and in that energy cloud.
Technically there should have been strong moral objections to those actions.
But they handwaved it with intact soul and memory explanations.
Cudos to Voyager for tackling this issue very well with Tuvix.
There they acknowledged that Tuvok and Neelix were in fact killed, a new person in his own right created then again killed and Tuvok and Neelix recreated.
So the transporter can do both, but usually the person stepping out of the transporter is the same person in every conceivable way.