What you are describing sounds a lot like a material constitution view of consciousness.
I think that, based on what I recall of Derek Parfit, there is a distinction that could be made between the continuity of consciousness and the connectivity. In the teletransporter problem, the continuity of your mind (or at least the body that constitutes your mind) is interrupted, and therefore your previous personal identity ends and a new one is created. However, the connectivity of your mind (relation R) is maintained, as your memory can still chart the path moment to moment as it went through the transporter.
This raises the question of which is more important, personal identity or relation R. Parfit argued that what really matters are your beliefs, memory, feelings, the things that make up your consciousness and not the body that "holds" it. Since these are maintained through relation R, you may not be "you" after going through the transporter, but everything important about you is still there.