Srila Prabhupada explains how the soul goes from one body to the next.

Srila Prabhupada: Yes. (indistinct) consider what kind of life I'm going to get. (indistinct) They have made their own rascal philosophy that there is no life after death. Although they're experiencing that I am changing my body in this duration of life. I know that I have a child's body, child's body. That body's finished. Still I'm existing. So why I shall not exist after this body's finished? Where is the logic? Where is the... The simple logic they cannot understand. So dull brain. And they're advanced in civilization, big, big professor, big, big Indologist, this logic. The simple reason they cannot understand. What did they say? You have talked to many men about this logic?
Devotee: About life after death?
Srila Prabhupada: Yes.
Devotee:  They say it doesn't go beyond... They have to agree that the baby's body is gone and the young man's body is gone. They have to agree. But they say that that doesn't mean logically that I have to take another body.
Srila Prabhupada: What is the other logic? If you have changed your body so many times, why not change this body? What is your reason? Natural course it should be that I have changed so many bodies so this body I shall change. This is natural logic. And what is his logic?
Devotee: So he said... They say it may be or it may not be.
Srila Prabhupada: But that is your rascaldom. But this is the real logic.
Devotee:: They also say that it is not... It is the same body, more or less...
Srila Prabhupada: Same body...
Devotee: But it is just developed, but it is not a different body.
Prabhupada: No. Developed means different body. Development means different body. They cannot say it is not different body. Then if it is not different, then go to childhood again. That means they're not human being. Human being means with logic. According to their definition, man is rational animal. They're not even rational. Like cats and dogs. There is no rationality. Cats and dogs also they have got rationality. Logic plus authority, Krsna says. How you can deny? That means they don't agree with Krsna's instruction. You see? This logic, I am not giving this logic. This logic is given by Krsna. So unless... The difficulty is that unless they accept the authority, it is very difficult. Logic is there. The authority is there.

Srila Prabhupada Morning Walk, June 8, 1974, Geneva, Switzerland

Srila Prabhupada explains how to understand the presence of the soul.

Devotee The example is that this body, there is a living force within the body which is causing it to move and to work, and that living force is the soul. And when the soul leaves this body, then the body is no longer moving and eating and talking and hearing and seeing. So in the same way in this universe there is a greater spiritual person, and His body, in a sense, is this whole universe, and it is moving on the same principle because He is the supreme consciousness.
Deshimaru: But what do you know about the soul leaving the body? Maybe when the body leaves the soul, the soul also dies.
Devotee:   The point is that body and soul are two things.
Deshimaru: I think it is same thing.
Srila Prabhupada: You think, but how you can prove?
Deshimaru: How you can prove that the body...?
Prabhupada: Yes, you cannot prove. When the soul is gone...
Devotee:   He's asking how can we prove our point?
Srila Prabhupada: Yes, I can prove that the body is dead because the soul is gone. (French)
Devotee:   He says, "Oh, I can also say that when the body is dead, the soul is dead."
Devotee:: No, but at the time of death the body is still there. Just like your body is there, at the time of death your body will still be there, but it won't be the same, something is missing. Something's gone away. What's the difference between a dead body and a living body?
Srila Prabhupada: How you can explain? Why the body's hand is there, leg is there, head is there, why it stopped working? (French)
Devotee:: He says that when there is no longer any movement in the cells, scientific explanation.
Srila Prabhupada: So why don't you... If you know that, then replace that cell.
Devotee:  The scientists don't have a perfect explanation either. They don't know exactly what death is.
Deshimaru: They cannot know also what life is.
Devotee: They do. They know this much, that life is what is not present in a dead body. So it must be different than the body.
Deshimaru: But the death is... You can change also what you say.
Srila Prabhupada: How you can change? Can you make the dead body alive? (French)
Devotee: I can say that life is not present in a dead body, but you can't say that death is present in a live body. (French)
Srila Prabhupada: No, he says that the living force and the body is the same.
Devotees: Yes.
Srila Prabhupada: Does he not say? But why the body becomes dead, and the living force go on? Why? Why the body is no more moving? (French)
Devotee:: He says when the body is dead, everything is gone, but what is still existing is the karma.
Srila Prabhupada: Is the karma.
Devotee:   The karma. What is still existing when a body is dead and when the soul is gone, the something is still there, still existing, is the karma.
Devotee:   Let's say if your hand is cut off, and it's lying there. Why is it that you are not conscious of that hand? (French)
Devotee: Because the nerves are just cut.
Srila Prabhupada: No, no, if the body and the hand is the same, when it is cut, then it is lying down on the floor. So why there is no consciousness. His question is very intelligent.
Deshimaru: Because the hand is cut.
Srila Prabhupada: But it is hand. Why there is no consciousness?
Devotee:  It's part of the body. It should have its own consciousness if the body and consciousness is the same.
Srila Prabhupada: You say the consciousness and body is the same. So when it is cut, why there is no consciousness? Therefore the body is different from consciousness. Therefore body is different from consciousness. (French)
Devotee: And also if he says that the soul and the body are the same, then what is the cause of death then? Then what is death?
Srila Prabhupada: No, we say the soul goes away from this body. Just like if I am sitting here, I can go away from this room, so similarly, the soul is within the body, he goes away, and therefore the body is dead.
Devotee: That's our explanation, but what is his explanation of death? If the soul and the body are the same then what is the cause of death.
Prabhupada: That he does not know, he cannot explain.

Srila Prabhupada Room Conversation with Mr. Deshimaru, June 13, 1974, Paris

Srila Prabhupada explains reincarnation.
Srila Prabhupada: So if a human being does not become inquisitive to understand what he is, in which way his progress should me made, then he remains a dog. The dog cannot do it. And we have got the capacity. If we neglect this facility and remain like a dog, simply engaged in eating, sleeping, sex, and defense, then we remain dog. Then again we become dog. The opportunity was given to us to understand the problems of life, how to solve. If you don't take this opportunity, facility, if you simply remain like dog, then we are next life... That also they do not understand, that there is next life. Do you believe in the next life? You, a person, do you believe in a next life?
Justin Murphy: No, I don't.
Srila Prabhupada: You do not.
Justin Murphy: No.
Srila Prabhupada: Just see. Now...
Justin Murphy: So therefore I'm bad material.
Srila Prabhupada: [break] ...believe it.
Justin Murphy: But I will become an old man anyway in the course of my life.
Srila Prabhupada: Yes. So therefore there is future life. If you say, "I don't believe in it,"...
Justin Murphy: Are you talking about reincarnation, life after death?
Srila Prabhupada: Yes. Reincarnation, you are already reincarnated. Where is your that childhood body? Where is that body?
Justin Murphy: Here it is. It's grown.
Srila Prabhupada: No. No, it is changed.
Justin Murphy: It's grown, it's changed, it's evolved, I have evolved. Just like evolution, I have evolved to the situations...
Srila Prabhupada: Anyway... Just try to understand. Anyway, that, your boy's body or childhood body is no longer. Either you say changed or grown, whatever you say, it doesn't matter.
Justin Murphy: But they're the same bones, it's the same skin.
Srila Prabhupada: But you are the same man. That's a fact. You understand that you were a child or you were a boy, youthful boy, jumping. You remember that body, but that body is not existing. That's a fact.
Justin Murphy: I can't agree.
Srila Prabhupada: And why not? Suppose somebody had seen your childhood body, and for many years he has not seen you, and he all of a sudden comes. Suppose your father's friend. So father introduces. He says, "Oh, you are the same?" He will be surprised because he saw you in a childhood body.
Justin Murphy: But I'm less interested in what...
Srila Prabhupada: No, no. First of all think that you have changed your body. The other man says, "Oh, you have grown up?" Or... Generally they take it as grown up. But the actual position is the body has changed.
Justin Murphy: But they're the same bones. It's the same skin. My face looks just about the same.
Srila Prabhupada: Not it is same. Medically, it is not the same.
Justin Murphy: The functions are different, but it's the same heart that's beating, the same veins...
Srila Prabhupada: No, no, no, it is not the same body. Just like in your childhood, when you were a boy, you had no sex impulse. Now you have got sex impulse. The body of a child, the body of a boy, they cannot understand sex life because the body is different. And now, because you have got different body, you can feel what is sex life. So it is imperceptibly changing. Therefore we think that it is growing. But it is changing. It is changing swiftly. Just like in the cinema spool. The picture is changing, but because it is changing so swiftly, you are seeing that one man is moving. That is the fact. There are hundreds and thousands of pictures passed on. When you see that "This man is taking the stick and bringing this way," this means there are many pictures. So similarly, it is like a spool. Your body is changing every moment. That is medical science.
Justin Murphy: Absolutely. Yes. At every moment.
Srila Prabhupada: So changing. So you are changing your body. That's a fact. But because you are seeing all in one spool, you are thinking, "It is growing; it is moving." That's all. But it is changing. This is the science. So body is changing. And you remember that you had such-and-such body. Therefore you are different from the body. This is the science. So unless we understand that "I am not this body. I am different from the body. I am changing bodies. Therefore I will have to change this body and accept another body..." This is the science, beginning of scientific knowledge. Without understanding this fact his advancement of knowledge is simply for eating, sleeping, sex, and defense. That's all. There is no advancement. According to Vedic literature, he remains animal. Sa eva go-kharaù [SB 10.84.13].

Srila Prabhupada Room Conversation with Mr. Justin Murphy (Geographer)
May 14, 1975, Perth, Australia