Based on the lifestory of Shakyamuni Buddha, one can see that there is an evolution, from disciple to boddhisatva until the Buddhahood stage... like a path his mind was following, but we know minds just go around samsara up and down, nothing except seeds of kamma goes along with the mind, no such thing as a soul. So how can one relate that to Anatta?
Karma does not go from life to life. Karma is the volitional action performed here and now. It only lasts a moment. But it can have future repercussions. The consequence is the result of a Karma. Not the Karma itself.
The question of if something goes from life to life would not even arise, if you understand what's happening here and now. Lets say now you are thinking about building a house in the future. If you think about it for an hour, does it mean your first thought about it lasted for an hour? No! The first thought moment appeared and vanished then and there. But as a consequence, similar thoughts kept appearing in your mind one after the other. But they all vanished immediately. You might think about the building of the house tomorrow or even months from now. But that doesn't mean your original thought lasted that long. It'll be a new thought, but it can be similar to the thoughts you had today.
So if nothing goes from this moment to the next, even in this life, how can anything go from life to life?
I think you've forgotten that anyone who wants to attain nibbana requires to develop the ten perfections (Paramis) in him in each life since the day he decides to go out of this vicious cycle of life and death.
- Generosity (Dāna)
- Morality (Sīla)
- Withdrawal (Nekkhamma)
- Understanding (Pañña)
- Enthusiastic Energy (Viriya)
- Patient Forbearance (Khanti)
- Honesty & Truthfulness (Sacca)
- Resolute Determination (Adhitthāna)
- Kind Friendliness (Mettā)
- Imperturbable Equanimity (Upekkhā)
The one who tries to become a Buddha must perfect them in three different ways.
Those who awaken as disciples = Savaka-Bodhis give all possessions away including wife and kids to perfect generosity. (Similar level with the 9 other perfections...)
Those who awaken as Solitary Buddhas = Pacceka-Buddhas give an organ, limb or eye away to perfect generosity. (Similar level with the 9 other perfections...)
Those who awaken as Perfect Buddhas = Sammāsam-Buddhas give even their own life away to perfect generosity. (Similar level with the 9 other perfections...)
So the 10 qualities he inculcates in each life will guide him as positive kamma along the way and will help him until he attains nibbana. As an example let's say you taught someone the dhamma (Dāna the first perfection), and as a result you may reap the benefit of it in the next life as being born among admirable friends (kalyanamitta).
The Buddha said:
"These few supreme mental qualities, will culminate in complete enlightenment. There is nothing elsewhere beyond them! Be thorough, firm and systematic so to complete them all..."