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)
buddhism.stackexchange.com/users/18/yuttadhammo
I've heard this issue described as one of the major philosophical problems for Buddhism.
How does rebirth make sense in the absence of a fixed self?
The problem is akin to the problem of evil for Christians - how does a good God allow bad things to happen.
And as fitting this kind of problem there's been a lot of historic opinion and arguably no definitive answer.
The thoughts and analogies that I have found useful have been One position is that there is rebirth but no-one is reborn.
I fit this satisfying mystical to be honest.
The analogy of the candle.
A new candle is lit from an old one and the flame passes from candle to candle.
We can't say it's the same flame
but then we can't say that the flames in the candles are unrelated.
The contention is that this is an instructive way to think about rebirth.
Another analogy of a pile of coins versus a string of beads.
Reincarnation (with a soul) can be viewed as a string of beads.
There is something (the thread/a soul) that runs through all beads.
A pile of coins is like rebirth (no soul).
There is nothing running in the pile of coins that gives them their location in the pile.
However it is the positions of all the previous coins that gives the uppermost coin it's elevated position.
If none of these work then there is always the Buddha's advice about n ot worrying about such metaphysical mechanics .
Knowing this stuff won't help you get liberated.
I am aware that Tibetan Buddhists will have a far more definite idea about how these things happen.
The perspective of this answer is probably more Theravadan but the book that I've sourced this from is more eclectic and even secular ( Exploring Karma and Rebirth by Nagapriya ).
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Looking at it from another angle, what would it mean to be reborn without your nature?
In what sense would the reborn being still be you?
Rebirth without persistence of nature is indistinguishable from no rebirth at all.
buddhism.stackexchange.com/users/737/Anonymous Coward
Everything is impermanent, even one's nature.
If you look at the span of your life, your year, your month, or even a single day in your life you will see that your nature is in a constant state of change.
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Agree with @Christopher Lee on this but with a small addition.
Some strong habits can persist over may lives and even after enlightenment, but in the long run they can also fade away.
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Nature doesn't move.
Your thinking is moving.
You cannot think about your nature, you cannot speak about it, because your thinking is a "product" made from the nature by your tendencies which again raise from your habits.
We need to perceive our nature to understand it.
Not to think about it.
It's like to think about meals instead to eat.
Your nature appears when you stop the mess in your head.
Right now!
Enlightenment
Helping others Congratulations! :-)
buddhism.stackexchange.com/users/740/Grumpa
Imagine a person that is very calm, generous, doesn't like to harm others etc. If this person gets a rebirth in the human realm will he/she develop the same characteristcs?   The point is that Kamma and the last mental state can go on, but according to Anatta I'm not sure about the rest.
Does the "nature" of a person move with her to the next life?
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buddhism.stackexchange.com/questions/2892
buddhism.stackexchange.com/users/533/konrad01
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. Some strong habits can persist over may lives and even after enlightenment, but in the long run they can also fade away.  Karma does not move on, as it is finished as soon as it is done. However, the effects of one moment have an effect on the next. Strong habits can persist over many lives and even after enlightenment. A person who ended a life with a certain nature will be reborn with a similar nature, due to the causal characteristic of reality. Whatever a bhikkhu ponders upon will become the inclination of his mind, with potential repercussions far into the future. However, everything is impermanent, even nature.
Karma does not move on, as it is finished as soon as it is done. However, the effects of one moment have an effect on the next, with future repercussions. Everything is impermanent, even nature. Strong habits can persist over many lives and even after enlightenment. A person who ended a life with a certain nature will be reborn with a similar nature, due to the causal characteristic of reality. Whatever a bhikkhu ponders upon will become the inclination of his mind, with potential repercussions far into the future.