Does the "nature" of a person move with her to the next life?
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.

&quot;Bhikkhus, whatever a bhikkhu frequently thinks and ponders upon, that will become the inclination of his mind.&quot;
MN 19 (Bodhi, trans)