If there be no soul or transmigrating entity that takes rebirth, who or what bears or enjoys the consequences or fruits of karma?

This is a very profound question that begs a profound answer. Karma is said to be based on Thoughts, wholesome or unwholesome. Science tells us that Thoughts, generate Energy and form the basis of Information. Science also tells us that Information can never be destroyed. David Tong, Professor of Theoretical Physics at Cambridge University in his discourse, ''Quantum Fields, the Real Building Blocks of the Universe" (utube) explains that even in a pure vacuum, there are phenomena that can never be destroyed ( minute 21 onwards). That seems to be the position of science today. Cosmologist and Physicist Jude Currivan (utube), is one of many modern thinkers, who advocate that "Information is at the Centre of Creation". This would imply that even if the Universe is destroyed, these phenomena will remain. I believe that these phenomena represent Information - Karma. Aeons of Information that can never be destroyed, also referred to as the Akashic record. According to Buddhist Cosmology, when a Universe is destroyed, the Karma of the Universe becomes the cause for a new Universe to be born. Karma is therefore a very profound subject and the Buddha advised us against probing deeply into it. Suggest you read "The Dhamma Theory" by Professor Y. Karunadasa (pdf) to get a better understanding of these phenomena. The Buddha suggested that we develop insight, through meditation, to realize the truth of his Doctrine of Anatta. That will surely end your Noble Quest to find the Truth.

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 & 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".

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?

I think it's based on observation -- e.g. if you stick your hand in the fire now, then in future the hand will be conditioned by its having been burned n the past.

Also I don't think that "the anatta theory" says that there's a "new individual". I think it says there's "no" individual, no perpetual/unchanging "self" -- e.g. "I" might be known as "son" early in life, perhaps known as "husband" or "co-worker" later in life, and maybe known by some other name entirely in future life.

Why the threat of subjecting another to suffering, if that being cannot be held responsible for actions in a previous life?

Thinking that, "this is me", and, "that will be another", is not anatta as far as I know.

They're examples of "self views" or "identity views" which we're advised to do away with -- e.g. because they result in "a thicket of views" i.e. confusion -- and result in suffering, if only because they're unsatisfactory -- and, I think, because they result from and perpetuate a sense of "attachment" and of "seeing the aggregates as self", i.e. the "clinging attachments" identified in the doctrine of the "four noble truths".

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?

Some people suggest that Buddhism was originally about liberation in this life, rather than about a fortunate rebirth in a future life.

Still, that was my first question on this site -- Is rebirth a delusional belief? -- for example ...

The Buddhist doctrine of "anatta" (there is no self?) and "anicca" (self is impermanent?) seem to me to be saying that, if (it is believed that) there is rebirth, that 'rebirth' is fairly meaningless, i.e. it is a rebirth of nothing in particular: why not just call it a "birth" instead of a rebirth?

... so you might want to read the answers to that.

It's a frequently-asked question.

I think I decided that the question may proceed from identity-view, which we're better off without, e.g. How is it wrong to believe that a self exists, or that it doesn't?

And I think that some suttas -- perhaps A Safe Bet (MN 60) -- might suggest that Buddhist practice is "right" (or "safe"), whether or not there is a future existence.

What is the transmigrating entity or soul experiencing moment by moment experience right now in you? There isn't any to be found. With some kind of good karma or good practice one can see one is made up of formations becoming born and dead every moment, there is no static soul, no static anything. Physical death is just another moment.

It's like the cloths you wore 10 years were referred to as 'your clothes', so the clothes you wear nowadays are referred to as 'your clothes' and the future clothes will be 'your clothes'; yet they are not the same clothes but are all called clothing and can be grasped with wrong view as self-referable.

Consciousness is to be understood(mn43), as according to the principle and function; arising as one ceasing as another, change is it's quality & characteristic.

Phenomena are mindmade and mindmade phenomena just come into play according to the principles of reason and are thought about as associated with this or that person.

The one who later enjoys / suffers the fruits of a prior action done by an agent is neither the same nor different than that agent

If you believe in karma, then it is not entirely different. If you believe in no-self, then not the same. Working out in what way that can be the case falls to dependent origination and theories of consciousness. The standard response is, I think, that only some mental events (the ones I label "me" vs the ones "you") occur in a causal series. How that can continue after death is perplexing.

1. here its clearly answered https://youtu.be/xjlBobj0iSA

2."Now at that moment this line of thinking appeared in the awareness of a certain monk: "So — form is not-self, feeling is not-self, perception is not-self, fabrications are not-self, consciousness is not-self. Then what self will be touched by the actions done by what is not-self?"

Then the Blessed One, realizing with his awareness the line of thinking in that monk's awareness, addressed the monks: "It's possible that a senseless person — immersed in ignorance, overcome with craving — might think that he could outsmart the Teacher's message in this way: 'So — form is not-self, feeling is not-self, perception is not-self, fabrications are not-self, consciousness is not-self. Then what self will be touched by the actions done by what is not-self?' Now, monks, haven't I trained you in counter-questioning with regard to this & that topic here & there?"

https://www.accesstoinsight.org/tipitaka/mn/mn.109.than.html

It is very important to understand how rebirth or birth takes place. There is ignorance, with ignorance as condition volitional formations come into existence , with volitional formations as condition consciousness comes into existence, with consciousness as condition name and form comes into existence, with name and form as condition six senses come into existence, with six senses as condition contact comes into play , with contact as a condition feelings come into play , with feelings as a condition cravings come into existence, with cravings as a condition clinging comes into play , with clinging as a condition attachments come into existence , with attachments as a condition becoming comes into play , with becoming as condition birth , ageing and death and whole mass of suffering comes into play.

This is how birth or rebirth occurs. Ignorance is at the beginning. What is ignorance? Ignorance is not knowing that birth , ageing, death is suffering… Ignorance is knowing that birth , ageing and death leads to happiness or gives some pleasure or is a divine purpose or solves any karmic problem ….

Now that I have told you how birth occurs , the answer should be clear to you.

According to Buddhism, a being is depedently arisen. For each moment to moment arisen state there is no unchanging core which is transmitted. In contemporary religions, there was a notion of an unchanging essence.

Past karmic action conditions over expences. With this as input and unripened karmic effects create the next moment of a being.

This question is asked and answered in the suttas. Two citations which might prove helpful:

From: AN 3.22.82-Woodward trans. http://buddhadust.net/dhamma-vinaya/pts/sn/03_kv/sn03.22.082.wood.pts.htm

"So then you say that body is not the Self;
feeling is not the Self,
perception is not the Self,
the activities are not the Self,
and consciousness is not the Self,

Then what self can those acts affect
which are not self-wrought?"

Thereupon the Exalted One,
with his thought reading the thoughts of that brother's mind,
said to the brethren:

"It is possible, brethren, that some senseless fellow,
sunk in ignorance
and led astray by craving,
may think to go beyond the Master's teaching thus:

'So then you say that body is not the Self;
feeling is not the Self,
perception is not the Self,
the activities are not the Self,
and consciousness is not the Self,

Then what self can those acts affect
which are not self-wrought?'

That question, brethren,
I have already answered thus and thus
in those teachings that I have given you.

From SN 2.12.17-Rhys Davids: http://buddhadust.net/dhamma-vinaya/pts/sn/02_nv/sn02.12.017.rhyc.pts.htm

"What then, Master Gotama,
is one's suffering wrought by another?

"Not so verily, Kassapa,"
said the Exalted One.

"What then, Master Gotama,
is suffering wrought both by one's self
and by another?"

"Not so verily, Kassapa,"
said the Exalted One.

"What then, Master Gotama,
has [the] suffering [which is] wrought neither by myself
nor by another,
befallen me by chance?"[5]

"Not so verily, Kassapa,"
said the Exalted One.

"What then, Master Gotama,
is suffering non-existent?"

"Nay, Kassapa, suffering is not non-existent;
suffering is."

"Then Master Gotama
neither knows nor sees suffering?"

"Nay, Kassapa, I am not one who knows not suffering
nor sees it.

I am one that knows suffering, Kassapa,
I am one that sees suffering."

"'One and the same person
both acts and experiences [the results]'

-this, Kassapa, which you called at first
'suffering self-wrought,'
amounts to the Eternalist theory.

'One acts, another experiences [the result]'

this, Kassapa, which
to one smitten by the feeling
occurs as

'suffering caused by another,'

amounts to the Annihilationist theory.

To you, Kassapa, the Tathāgata,
not approaching either extreme,
teaches the Norm by a middle [way]:

Conditioned by ignorance activities come to pass,
conditioned by activities consciousness;
conditioned by consciousness, name-and-shape;
conditioned by name-and-shape, sense;
conditioned by sense, contact;
conditioned by contact, feeling;
conditioned by feeling, craving;
conditioned by craving, grasping;
conditioned by grasping, becoming;
conditioned by becoming,
birth,
decay-and-death,
grief,
suffering,
sorrow,
despair come to pass.

But from the utter fading away and ceasing of ignorance [comes] ceasing of activities;
from ceasing of activities ceasing of consciousness;
from ceasing of consciousness ceasing of name-and-shape;
from ceasing of name-and-shape ceasing of sense;
from ceasing of sense ceasing of contact;
from ceasing of contact ceasing of feeling;
from ceasing of feeling ceasing of craving;
from ceasing of craving ceasing of grasping;
from ceasing of grasping ceasing of becoming;
from ceasing of becoming ceasing of birth;
from ceasing of birth,
old age-and-death, grief, lamenting, suffering, sorrow, despair cease.

Such is the ceasing of this entire mass of ill."