Through which medium does kammic information travel after the break-up of the body?
Questions of King Milinda say, "it is nama-rupa that is reborn". When the King asks whether it is the same nama-rupa, Nagasena says, it is not the same, but this nama-rupa arises in causal continuity with the previous nama-rupa, and thus we talk about "rebirth".

Questions of King Milinda also compare rebirth to a memorized verse passing from one person to another, and to a flame passing from one lamp to another. It is clear that rebirth is about causal continuity and the passage of (some) information.

Japanese Zen patriarch Dogen explained that the underlying nature of mind is not separate from the underlying nature of the rest of material world. In numerous explanation in Zen and Vajrayana there is a rather clear consensus that the nature of mind are dynamically interacting relationships, both inside the person, and between the parts of the world.

So when we are talking about rebirth we are talking about continuity of interacting relationships implicitly "stored" or "encoded" in the fabric of activities happening in the world at large.

So the media that carries the karma after the break-up of the body is the world itself. People, things, activities, relationships carry the imprints of the previous existence into the next.

At least, this is my understanding.