If there be no soul or transmigrating entity that takes rebirth, who or what bears or enjoys the consequences or fruits of karma? How can a religion or school of thought justify or rationalise the proposal that potential suffering could be inflicted on a subsequent rebirth - to all intents and purposes, a new individual, according to the anatta theory - because of the actions of a previous individual?
Why the threat of subjecting another to suffering, if that being cannot be held responsible for actions in a previous life?
If no soul or persistence exists, how can one suggest that performing positive karmic actions will improve the quality of life of a subsequent rebirth, if there is no element retained of a previous iteration to reap the reward and receive the incentive to continue to perform good karmic actions and/or continue in Buddhist schooling?
The question is irrelevant because there are two levels of teaching in Buddhism (refer to two sorts of right view in MN 117). The Buddha generally did not teach about "kamma-vipaka" and "not-self" (anatta) together. 

"Kamma-vipaka" is mainly for puthujjana (common village people who need to be taught morals so to avoid self-harm) and "anatta" is for Noble People. Only later heretics (such as the Brahman philosopher Buddhaghosa) taught "empty not-self phenomena are reborn in a Twelvefold Voidness". 

When a Noble Person realises the reality of not-self (anatta), they no longer perform immoral actions because there is no longer selfishness or self-interest within their mind. Also, there is no self-referent to any past kamma. Refer to AN 6.63, which says: 


  Just this noble eightfold path — right view, right resolve, right speech, right action, right livelihood, right effort, right
  mindfulness, right concentration — is the path of practice leading to
  the cessation of kamma.


But for the common village people who have not realised not-self (anatta), the Buddha taught they themselves are the owners &amp; heirs of their personal actions, as follows: 


  Bhikkhus, beings are the owners of their kamma, the heirs of their kamma; they have kamma as their origin, kamma as their relative, kamma
  as their resort; whatever kamma they do, good or bad, they are its
  heirs.
  
  AN 10.216


The later is true because when kamma-vipaka occurs, the idea of "self" arises in the mind of the recipient of kamma, such as the regret: "Why did I do that bad deed? What did I do to myself? I have ruined my life". 

In summary, whenever "rebirth" occurs, it is a "rebirth" of "self". Inwardly, it is never "not-self".