"By self is one defiled,
By self is one purified."
Is it the doer of the act or another who reaps its results in the succeeding birth? 
To say that he who sows is absolutely the same as he who reaps is one extreme, and to say that he who sows is totally different from he who reaps is the other extreme. Avoiding these two extremes, the Buddha teaches the doctrine of the middle way in terms of cause and effect. "Neither the same nor another" (na ca so na ca a, writes the Venerable Buddhaghosa in the Visuddhi Magga. The evolution of the butterfly may be cited in illustration.
Its initial stage was an egg. Then it turned into a caterpillar. Later it developed into a chrysalis, and eventually into a butterfly. This process occurs in the course of one lifetime. The butterfly is neither the same as, nor totally different from, the caterpillar. Here also there is a flux of life, or a continuity.
Venerable Nāgasena explains this point by citing the illustration of a lamp that burns throughout the night. The flame of the first watch is not identical with that of the last watch, yet throughout the night the light burns in dependence upon one and the same lamp. As with the flame so there is a continuity of life -- each succeeding stage depending upon the preceding one.
If there be no soul, can there be any moral responsibility? 
Yes, because there is a continuity or identity in process, which is substituted for an identical personality.
A child, for instance, becomes a man. The latter is neither absolutely the same as the former -- since the cells have undergone a complete change nor totally different -- being the identical stream of life. Nevertheless, the individual, as man, is responsible for whatever he has done in his childhood. Whether the flux dies here and is reborn elsewhere, or continues to exist in the same life, the essential factor is this continuity. Suppose a person was 'A' in his last birth, and is 'B' in this. With the death of 'A' the physical vehicle, the outward manifestation of Kammic energy is relinquished and, with the birth of 'B' a fresh physical vehicle arises. Despite the apparent material changes, the invisible stream of consciousness (cittasantati) continues to flow, uninterrupted by death, carrying along with it all the impressions received from the tributary streams of sense. Conventionally speaking, must not 'B' be responsible for the actions of 'A' who was his predecessor?
Some may object that in this case there is no memory owing to the intervening death.
But is identity or memory absolutely essential in assessing moral responsibility?
Strictly speaking, neither is essential.
If, for instance, a person were to commit a crime and suddenly, losing his memory, were to forget the incident, would he not be responsible for his act?
His forgetfulness would not exempt him from responsibility for the commission of that crime. To this, some may ask: "What is the use of punishing him, for he is not aware that he is being punished for that crime? Is there any justice here?"
Of course, there is not, if we are arbitrarily governed by a God who rewards and punishes us.
Buddhists believe in a just and rational law of Kamma that operates automatically and speak in terms of cause and effect instead of rewards and punishments.
In the words of Bhikkhu Silācāra:
"If a person does something in sleep, gets out of bed and walks over the edge of a verandah, he will fall into the road below and in all likelihood break an arm or leg. But this will happen not at all as a punishment for sleep-walking, but merely as its result. And the fact that he did not remember going out on the verandah would not make the slightest difference to the result of his fall from it, in the shape of broken bones. So the follower of the Buddha takes measures to see that he does not walk over verandahs or other dangerous places, asleep or awake, so as to avoid hurting himself or anyone who might be below and on whom he might fall."
The fact that a person does not remember his past is no hindrance to the intelligent understanding of the working of Kammic law. It is the knowledge of the inevitability of the sequence of Kamma in the course of one's life in Samsāra that helps to mould the character of a Buddhist.