We Buddhists from the past and up to present day has faced some difficulty to reconcile the anatta doctrine and how karma and rebirth works without any sense of self. While other religious teachings will easily explain that it is the soul/self/conciousness which performs action and migrates from one life to another, Buddhists have to take great pains to explain the process of rebirth without a migrating self because not everyone can understand the concept of dependent origination (paticcasamuppada).
So Buddhist schools develop more specific terms and explanations on how the process of karma and rebirth which does not involve the migrating self. Beside from Pudgalavada, we can list some of them here:
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Sarvastivada (Vaibhasika) school with the doctrine of “all exists” (sarvam asti: all phenomena exist in the past, present and future) explains that when actions (karma) are performed, a phenomenon (dharma) called prapti (possession) will be formed, which exists forever until it gives consequences in the future. This prapti is a force that keeps karma from wrongly giving its fruit to the different doer. With this concept, there is no need for the self to be the doer and recipient of karmic consequences.
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Mahasanghika school has the concept of root consciousness (mulavijnana), the basis of consciousness on which karma and rebirth happens based on the interpretation of the term “luminous mind” (pabhassara citta) in the Pabhassara Sutta (AN 1.49–50)
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Sautrantika school which rejects the concept of “all exists” (and prapti) from Sarvastivada Vaibhasika above states that when an action (karma) in form of intention (cetana) is performed, it immediately disappears, but it leaves a trace/impression (vasana) in the stream of consciousness (cittasamtana) and experiences development/evolution (viparinama) over time until it finally gives its results. This concept is also called karmabija (seed of karma).
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Theravada school explains the process of karma and rebirth with the concept of bhavanga (I don’t have to explain it here because it is a popular Buddhist explanation which you can find in Abhidhamma by googling it).
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Mahayana Yogacara (Vijnanavada) school explains the process of rebirth and karma with the concept of alayavijnana, i.e the eighth consciousness (the other 7 consciousnesses are the five sense consciousnesses and mind consciousness [manovijnana] and the mind [mano] itself) in which all the seeds of good kamma and bad (karmabija) and all potential stain and purity of mind are stored. (You can also google it for more information).
All these schools claim that their explanation is in line with the Buddha teaching of anatta, all these terms are not self according to the respective school, including the Pudgalavadins. And the Pudgalavada school was very popular in their time and was the largest Buddhist school with the largest number of followers among all other Buddhist schools in India according to Xuanzang’s travel record [cmiiw] (perhaps because their concept was more easily accepted by lay people who were not engaged in sophisticated doctrines like Abhidhamma/Abhidharma and so on).
So, all these schools explanation are right in their own. They are just trying to make sense the anatta doctrine to their fellow Buddhists and non-Buddhists, including nowadays Theravadin Ajahns who explain it with their own terms and understanding.