kamma doesn't actually go on. kamma means action, and it is finished after it is performed. Nothing actually goes on from one moment to the next; the effects of one moment are felt in the next, and have potential repercussions far into the future.
It is these repercussions that do indeed create a sense of continuity from life to life; a person who ended their life with a certain nature will be reborn with a similar nature, simply due to the causal nature of reality.
"Bhikkhus, whatever a bhikkhu frequently thinks and ponders upon, that will become the inclination of his mind."
MN 19 (Bodhi, trans)