As stated in Bhagavad-gītā (2.20), the living entity is never created (na jāyate mriyate vā kadācin). There are five gross material elements and three minor material elements (mind, intelligence and ego), and there are eternal living entities. The living entity desires a certain type of body, and by the order of the Supreme Personality of Godhead, that body is created from material nature, which is nothing but a kind of machine handled by the Supreme Lord. The Lord gives the living entity a particular type of mechanical body, and the living entity must work with it according to the law of fruitive activities. Fruitive activities are described in this verse: karma-pamanīṁ māyām. The living entity is seated on a machine (the body), and according to the order of the Supreme Lord, he operates the machine. This is the secret of transmigration of the soul from one body to another. The living entity thus becomes entangled in fruitive activities in this material world. As confirmed in Bhagavad-gītā (15.7), manaḥ ṣaṣṭhānīndriyāṇi prakṛti-sthāni karṣati: the living entity is struggling very hard against the six senses, which include the mind.
(Srimad Bhagavatam----5:17:24----purport).

According to the Ṛg Veda, if at the time of sex the secretions of the mother are more profuse than those of the father, the child will receive a female body, and if the secretions of the father are more profuse than those of the mother, the child will receive a male body. These are the subtle laws of nature, which act according to the desire of the living entity. If a human being is taught to change his subtle body by developing a consciousness of Kṛṣṇa, at the time of death the subtle body will create a gross body in which he will be a devotee of Kṛṣṇa, or if he is still more perfect, he will not take another material body but will immediately get a spiritual body and thus return home, back to Godhead. This is the process of the transmigration of the soul. Therefore instead of trying to unite human society through pacts for sense gratification that can never be achieved, it is clearly desirable to teach people how to become Kṛṣṇa conscious and return home, back to Godhead. This is true now and, indeed, at any time.
(Srimad Bhagavatam----6:1:54----purport).

Transmigration of the soul is very clearly explained in this verse. One sometimes forgets his present body and thinks of his childhood body, a body of the past, and of how one was playing, jumping, talking and so on. When the material body is no long workable, it becomes dust: "For dust thou art, and unto dust shalt thou return." But when the body again mixes with the five material elements—earth, water, fire, air and ether—the mind continues to work. The mind is the subtle substance in which the body is created, as we actually experience in our dreams and also when we are awake in contemplation. One must understand that the process of mental speculation develops a new type of body that does not actually exist. If one can understand the nature of the mind (manorathena) and its thinking, feeling and willing, one can very easily understand how from the mind different types of bodies develop.
(Srimad Bhagavatam----10:1:41----purport).

One's consciousness, therefore, must always be absorbed in Kṛṣṇa, and then one's life will be successful. Otherwise the mind will carry the soul to another material body. The soul will be placed in the semen of a father and discharged into the womb of a mother. The semen and ovum create a particular type of body according to the form of the father and mother, and when the body is mature, the soul emerges in that body and begins a new life. This is the process of transmigration of the soul from one body to another (tathā dehāntara-prāptiḥ (B.G.2:13). Unfortunately, those who are less intelligent think that when the body disappears, everything is finished. The entire world is being misled by such fools and rascals. But as stated in Bhagavad-gītā (2.20), na hanyate hanyamāne śarīre. The soul does not die when the body is destroyed. Rather, the soul takes on another body.
(Srimad Bhagavatam----10:1:41----purport).

When there is a fire in a village, the fire sometimes jumps over one house and burns another. Similarly, when there is a forest fire, the fire sometimes jumps over one tree and catches another. Why this happens, no one can say. One may set forth some imaginary reason why the nearest tree or house did not catch fire whereas a tree or house in a distant place did, but actually the reason is destiny. This reason also applies to the transmigration of the soul, by which a prime minister in one life may become a dog in the next. The work of unseen destiny cannot be ascertained by practical experimental knowledge, and therefore one must be satisfied by reasoning that everything is done by supreme providence.
(Srimad Bhagavatam----10:1:51----purport).

"As the embodied soul continuously passes, in this body, from boyhood to youth to old age, the soul similarly passes into another body at death. The self-realized soul is not bewildered by such a change." Such verses from the Bhagavad-gītā or any other Vedic literature give valuable instructions on the occasion of someone's passing away. By discussing such instructions from the Bhagavad-gītā or Śrīmad-Bhāgavatam, a sober man can certainly understand that the soul never dies but rather passes from one body to another. This is called transmigration of the soul. A soul comes into this material world and creates bodily relationships with a father, a mother, sisters, brothers, a wife and children, but all these relationships pertain to the body, not the soul. Therefore, as described in the Bhagavad-gītā, dhīras tatra na muhyati: one who is sober is not disturbed by such phenomenal changes within this material world. Such instructions are called tattva-kathā, or real truth.
(Sri Caitanya Caritamrta----1:16:23----purport).

Just as we find different forms, we find different states of consciousness. When we see different states of consciousness, we may take it for granted that the bodies are different. In other words, different types of bodies depend on different states of consciousness. This is also confirmed in the Bhagavad-gītā (8.6):
yaṁ yaṁ vāpi smaran bhāvaṁ tyajanty ante kalevaram
taṁ tam evaiti kaunteya sadā tad bhāva-bhāvitaḥ
“One's consciousness at the time of death determines one's type of body in the next life.” This is the process of transmigration of the soul. A variety of bodies is already there; we change from one body to another in terms of our consciousness.
(Sri Caitanya Caritamrta----2:19:138----purport).