Generally, the 4th Way does not emphasize the concept of reincarnation. For the common man, reincarnation is not seen as specially central. Some exceptional persons incarnate into the world for a specific mission. In order to be able to do so, one must already have achieved essentially all spiritual development possible to the human form.
For the rest of humanity, Gurdjieff does not really speak of anything recognizable surviving physical death. He says blessed is he who has no soul and blessed is the one who has one, but woe to one who has one in embryo. Here we understand soul as meaning something that can survive the death of body while staying a recognizable unit and can possibly consciously reincarnate. This is however said to be very rare.
For half-crystallized fragments of souls, Gurdjieff describes these as wandering in the planetary sphere and incarnating as the chance may occur in whatever life forms may be available. Ones without any significant crystallization sort of melt into a pool of indifferentiated energies.
Ouspensky and Mouravieff write about recurrence. Recurrence is a mechanical replay of mechanical circumstances. This is not exactly reincarnation, although this too involves a person having repeated mechanical lives.
Recurrences take place in eternity. Eternity does not here mean an endless stretch of linear time but rather the set of all possible 'parallel' realities. In these parallel variants, countless essentially similar recurrences of one life can play themselves out ad infinitum. Eternity can be seen as a distinct dimension orthogonal to linear time.
Conscious influences can enter these lives from another plane, as it were. These are so-called B and C influences. These may be works of objective art, books, teachings, esoteric schools and so forth and represent a possibility for escaping mechanical recurrence. By recognizing such influences, one can increase consciousness in one's life and gradually come under another law, the Law of Exception. This in itself does not guarantee one's escape but can affect another round of repetition by giving one increased capabilities or motivation for more work towards consciousness.
Mouravieff uses the term film when speaking of the recurring pattern of a life, mechanically proceeding on parallel tracks in the dimension of eternity. Certain films may have an esoteric purpose. It is the task of the people in incarnation to find their respective cast members and perform a certain task together. The films play themselves in the dimension of eternity, potentially perpetually repeating but If one seeks to awaken and follow B influences the film may change and possibly lead to liberation from mechanical recurrence.
Again, one round will not be enough but there may be progress between rounds or on the other hand one may reject conscious influences in which case they may not reappear.
Ouspensky's book 'The Strange Life of Ivan Ousakan' illustrates the concept. Ivan is disillusioned with life, love and the world, in other words is facing a personal 'moral bankruptcy.' He goes to the magician and asks to be transported to his childhood with all memories intact, so that he may have another try. This happens and he makes the very same mistakes, his knowledge does not help him and on the contrary he forgets gradually how he returned and becomes again identified with a rerun of the same life. He faces the same quandary again and goes to the magician. This time he catches himself and instead of asking to go back yet again he asks to become a student of the magician, has met C influence. In principle one can repeat life endlessly but since the magician comes from the conscious realm, finding him again in the world of mechanical influences is in no way guaranteed.
Generally, the 4th Way teaching about reincarnation or recurrence is not very explicit nor is it central to the Work since one can only do Work in the present.
Gurdjieff's reference to 'he who has no soul' above may or may not have referred to 'preadamic man' as Mouravieff uses the term. Generally, the concept is not found in Gurdjieff's teaching.