Of all the esoteric traditions of humanity, only two speak specifically and in detail of a Divine Soul or spiritual principle that specifically evolves and grows through successive lifetimes and reincarnations. These are the Theosophy of Madame Blavatsky and its many orthodox and heterodox offshoots, and the Integral Yoga and Philosophy of Sri Aurobindo (who indicates he was inspired or shown this by the Mother (Mirra). I originally thought this meant that both derived this insight from Theon (who was Mirra's teacher in occulktism and who, she claimed, also was part of an early occult group with Blavatsky). But it seems that at least in his Hermetic Brotherhood of Luxor and Cosmic Review days, Theon denied the existence of reincarnation (although it is of course possible he adopted it later, but this would not indicate a link with Blavatsky)
Although it is true that earlier versions of this insight can be found in Hinduism (the soul passes through 84 crores (100,000 years) of incarnations, occupying succesively higher bodies) and there are occasional poetic references elsewhere, e.g. in Rumi, in none of these is the understanding specifically developed. In Hinduism for example if in the brief period the soul occupies a human form one doesn't use the opportunity to attain enlightenment, one then falls back into lower forms of existence for millions of lifetimes, until the cycle repeats itself. Only Meher Baba, who's position here and inmany other respects is quite original, seems to have a single cycle of once-only involution and evolution.
The following detailed description by Sri Aurobindo of the nature and growth of the Soul or Psychic Being is remarkably similar to the position of Theosophy, both i its Blavatskyian and later (Adyar Theosophy, Alice Bailey, current New Age, etc) forms. Sri Aurobindo had read Blavatsky (he mentions her a few times in his Letters, and even occasionally uses some of her terminology), so it is not impossible that there is some borrowing there, although the influence of Mirra (who was an accomplished occultist) is probably much stronger.
"the psychic part within is there to support the natural evolution, and the first natural evolution must be the development of body, life and mind, successively, and these must act each in its own kind or together in their ill-assorted partnership in order to grow and have experience and evolve. The soul gathers the essence of all our mental, vital and bodily experience and assimilates it for the farther evolution of our existence in Nature; but this action is occult and not obtruded on the surface. In the early material and vital stages of the evolution of being there is indeed no consciousness of soul; there are psychic activities, but the instrumentation, the form of these activities are vital and physical, - or mental when the mind is active. For even the mind, so long as it is primitive or is developed but still too external, does not recognise their deeper character. It is easy to regard ourselves as physical beings or beings of life or mental beings using life and body and to ignore the existence of the soul altogether: for the only definite idea that we have of the soul is of something that survives the death of our bodies; but what this is we do not know because even if we are conscious sometimes of its presence, we are not normally conscious of its distinct reality nor do we feel clearly its direct action in our nature..
As the evolution proceeds, Nature begins slowly and tentatively to manifest our occult parts; she leads us to look more and more within ourselves or sets out to initiate more clearly recognisable intimations and formations of them on the surface. The soul in us, the psychic principle, has already begun to take secret form; it puts forward and develops a soul-personality, a distinct psychic being to represent it. This psychic being remains still behind the veil in our subliminal part, like the true mental, the true vital or the true or subtle physical being within us: but, like them, it acts on the surface life by the influences and intimations it throws up upon that surface; these form part of the surface aggregate which is the conglomerate effect of the inner influences and upsurgings, the visible formation and superstructure which we ordinarily experience and think of as ourselves. On this ignorant surface we become dimly aware of something that can be called a soul as distinct from mind, life or body; we feel it not only as our mental idea or vague instinct of ourselves, but as a sensible influence in our life and character and action. A certain sensitive feeling for all that is true and good and beautiful, fine and pure and noble, a response to it, a demand for it, a pressure on mind and life to accept and formulate it in our thought, feelings, conduct, character is the most usually recognised, the most general and characteristic, though not the sole sign of this influence of the psyche. Of the man who has not this element in him or does not respond at all to this urge, we say that he has no soul. For it is this influence that we can most easily recognise as a finer or even a diviner part in us and the most powerful for the slow turning towards some aim at perfection in our nature.
But this psychic influence or action does not come up to the surface quite pure or does not remain distinct in its purity; if it did, we would be able to distinguish clearly the soul element in us and follow consciously and fully its dictates...
Moreover, the psychic being, the soul-personality in us, does not emerge full-grown and luminous; it evolves, passes through a slow development and formation; its figure of being may be at first indistinct and may afterwards remain for a long time weak and undeveloped, not impure but imperfect: for it rests its formation, its dynamic self-building on the power of soul that has been actually and more or less successfully, against the resistance of the Ignorance and Inconscience, put forth in the evolution upon the surface. Its appearance is the sign of a soul-emergence in Nature, and if that emergence is as yet small and defective, the psychic personality also will be stunted or feeble. It is too, by the obscurity of our consciousness, separated from its inner reality, in imperfect communication with its own source in the depths of the being; for the road is as yet ill-built, easily obstructed, the wires often cut or crowded with communications of another kind and proceeding from another origin: its power to impress what it receives upon the outer instruments is also imperfect; in its penury it has for most things to rely on these instruments and it forms its push to expression and action on their data and not solely on the unerring perceptions of the psychic entity. In these conditions it cannot prevent the true psychic light from being diminished or distorted in the mind into a mere idea or opinion, the psychic feeling in the heart into a fallible emotion or mere sentiment, the psychic will to action in the life-parts into a blind vital enthusiasm or a fervid excitement: it even accepts these mistranslations for want of something better and tries to fulfil itself through them. For it is part of the work of the soul to influence mind and heart and vital being and turn their ideas, feelings, enthusiasms, dynamisms in the direction of what is divine and luminous; but this has to be done at first imperfectly, slowly and with a mixture. As the psychic personality grows stronger, it begins to increase its communion with the psychic entity behind it and improve its communications with the surface: it can transmit its intimations to the mind and heart and life with a greater purity and force; for it is more able to exercise a strong control and react against false mixtures; now more and more it makes itself distinctly felt as a power in the nature. But even so this evolution would be slow and long if left solely to the difficult automatic action of the evolutionary Energy; it is only when man awakes to the knowledge of the soul and feels a need to bring it to the front and make it the master of his life and action that a quicker conscious method of evolution intervenes and a psychic transformation becomes possible."