Did Pudgalavada Buddhists claim the the "person" existed between lives and / or in one life?

I'm not very familiar with all the technical terms or schools things, but many of the Early Buddhist Schools original works are preserved in the Chinese Canon, such as, doctrine of Vibhajyavada sect should be found.

But to another half of your title question:

...the “person” existed between lives and / or in one life?

Xuanzang's (玄奘 602–664CE) Karika on the Eight Consciousness has this piece: 去後來先作主翁. Means, the Alaya the last leaving the dying body and the first entering the concoction when life conceived. However, in other Sutras, the Buddha did stress that he hesitated to teach about this higher teaching for those unfitted would mistakenly grasping it as the Self, he only taught it to the qualified. Thus, Alaya shouldn't be simply equated as a "person", in this sense.

Nagarjuna clearly analyzed in his Madhyamaka, that saying there is Self or there isn't Self both are wrong.

There is huge misunderstanding about "mind stream" in convention Buddhist scholastic discussion; related to the Anatman/Anatta notion. Thus it inevitably caused awkward and unnecessary argument about if there is rebirth or not. This is mainly due to the incomplete understanding by referring to the incomplete teachings. If, those scholars or conventional Buddhists have the access to the Mahayana Canon, especially the Chinese, they shouldn't have this doubt. The mind stream never ceases, whether live or dead. Yet the mind stream also never is a stream, like a stream filled fully with water. It's more like the blinking light-bulbs on the X'mas tree, because they are blinking so fast, it gives you an illusion the lights trailing from end to end, like a stream of light. This is my metaphor why it appeared there a person. One is able to understand properly if one read and got the meaning from the higher teachings preserved in Mahayana Sutras.

I would not go as far as to call Pudgalavada "a school", as with most early Buddhism there were people who held certain position, that later received a name and was used for comparison with mainstream traditions.

If I remember correctly, Pudgalavadins claimed that person exists for all practical intents and purposes, so I think that means conventional existence, not ultimate. You should be able to find more details on Pudgalavada in Ch. 9 of Abhidharmakosa-Bhasya of Vasubandhu. See this for a quick summary - pp. 89-91:

The Pudgalavadins introduce the notion that persons are conceived on the basis of the skandhas as fire is conceived in reliance upon fuel without being other than or the same in existence as fuel. Pudgalavadins assert that in addition to phenomena that are conditioned & unconditioned (samskrta/asamskrta), impermanent and permanent (anitya/nitya), the 12 ayatanas, the 18 dhatus, etc., there are, in addition, phenomena that are inexplicable (avakrtavya). For the Pudgalavadins, conventional realities may be substantially established (as they are for the Vaibhasikas) or inexplicable. Inexplicable entities are entities without separate identities – they are neither the same as, nor different from, the skandhas. The Pudgalavadins reject the logic that everything is either substantially real or substantially established. Inexplicable persons are conventional realities insofar as they are conceived in dependence upon the skandhas, but ultimately exist insofar as they exist apart from being conceived. Persons are single entities without separate identities, an inexplicable unity.
The Pudgalavadins maintain that the person is perceived through an inexplicable perception incidental to a consciousness perceiving its proper object. It is an inexplicable perception because it is neither the same nor different from the perception of the object.
The basic attitude of Vasubandhu, and others, toward the Pudgalavadins is that in their theory of persons, it can appear that the person is substantially real, despite attempts by the Pudgalavadins to refute this attribution.

Mahayana, in particular Tibetan Nyingma tradition, insists that to say that person conventionally exists still leaves the mind grasping after the person, so more for didactic than for analytical purposes they say that there is no person is the right view. See "Mipham's Beacon of Certainty" (Mipham uses the traditional rhetoric of "snake", "vase" etc. but what he means is person, or entity in the broad sense)

Also see Satyasiddhisastra of Harivarman pages 67-74 for various other views around Pudgala and their refutations. As of late I consider Satyasiddhisastra a key text providing the missing link between Early Buddhism and Mahayana, with Abhidharmakosa and Mulamadhyamakakarika correspondingly on one and the other sides of the divide (conceptually, not chronologically).