Rebirth as cycle of consciousness?

In Theravada, these questions are considered abhidhamma material. The goto reference is the Comprehensive Manual of Abhidhamma, which says in regards to your second question:

2. Is consciousness inevitably always a sequence of moments, not a continuum of awareness? Is this also true during meditation?

A second distinguishing feature of the Abhidhamma is the dissection of the apparently continuous stream of consciousness into a succession of discrete evanescent cognitive events called cittas, each a complex unity involving consciousness itself, as the basic awareness of an object, and a constellation of mental factors (cetasika) exercising more specialized tasks in the act of cognition.

In the Theravada, reality is reality, whether you are meditating or not.

As to the first set of questions:

Can you give any citations for this explanation of 'rebirth'?

Mahasi Sayadaw says:

Before death the stream of consciousness depends on the physical body and is continuous with one unit following the other uninterruptedly. After death, the body disintegrates and the stream of consciousness shifts to the physical process in another abode. This may be likened to the continuous appearance of light in an electric bulb through the ceaseless generation of electricity. When the bulb is burnt up, the light goes out but the potential electric energy keeps on coming. Light reappears when the old bulb is replaced with a new one. Here, the bulb, energy and light are all changing physical processes and we should be mindful of their impermanent character.

(source)

Which tradition[s] of Buddhism is this from?

Maybe better to ask which it is compatible with... it is compatible with Theravada, and probably most other orthodox schools.

Is it described in any canonical suttas

As with many doctrines, the concept of rebirth as a continuation of a momentary string of consciousness is not explicitly spelled out in the Suttas. The way rebirth works is implicit, for example:

“Bhikkhus, what one intends, and what one plans, and whatever one has a tendency towards: this becomes a basis for the maintenance of consciousness. When there is a basis there is a support for the establishing of consciousness. When consciousness is established and has come to growth, there is the production of future renewed existence. When there is the production of future renewed existence, future birth, aging-and-death, sorrow, lamentation, pain, displeasure, and despair come to be. Such is the origin of this whole mass of suffering.

-- SN 12.38 (Bodhi, trans)

“Bhikkhus, though someone might say: ‘Apart from form, apart from feeling, apart from perception, apart from volitional formations, I will make known the coming and going of consciousness, its passing away and rebirth, its growth, increase, and expansion’—that is impossible.

3“Bhikkhus, if a bhikkhu has abandoned lust for the form element, with the abandoning of lust the basis is cut off: there is no support for the establishing of consciousness.

If he has abandoned lust for the feeling element … for the perception element … for the volitional formations element … for the consciousness element, with the abandoning of lust the basis is cut off: there is no support for the establishing of consciousness.

-- SN 22.53 (Bodhi, trans)

“Sisters, suppose an oil-lamp is burning: its oil is impermanent and subject to change, its wick is impermanent and subject to change, its flame is impermanent and subject to change, and its radiance is impermanent and subject to change. Now would anyone be speaking rightly who spoke thus: ‘While this oil-lamp is burning, its oil, wick, and flame are impermanent and subject to change, but its radiance is permanent, everlasting, eternal, not subject to change’?”

“No, venerable sir. Why is that? Because, venerable sir, while that oil-lamp is burning, its oil, wick, and flame are impermanent and subject to change, so its radiance must be impermanent and subject to change.”

“So too, sisters, would anyone be speaking rightly who spoke thus: ‘These six internal bases are impermanent and subject to change, but the pleasant, painful, or neither-painful-nor-pleasant feeling that one experiences in dependence upon the six internal bases is permanent, everlasting, eternal, not subject to change’?”

“No, venerable sir. Why is that? Because each feeling arises in dependence upon its corresponding condition, and with the cessation of its corresponding condition, the feeling ceases.”

“Good, good, sisters! So it is with a noble disciple who sees this as it actually is with proper wisdom.

-- MN 146 (Bodhi, trans)

5“Master Gotama, when a flame is flung by the wind and goes some distance, what does Master Gotama declare to be its fuel on that occasion?”

6“When, Vaccha, a flame is flung by the wind and goes some distance, I declare that it is fuelled by the wind. For on that occasion the wind is its fuel.”

7“And, Master Gotama, when a being has laid down this body but has not yet been reborn in another body, what does Master Gotama declare to be its fuel on that occasion?”

8“When, Vaccha, a being has laid down this body but has not yet been reborn in another body, I declare that it is fuelled by craving. For on that occasion craving is its fuel.”

-- SN 44.9 (Bodhi, trans)

When this was said, the Venerable Moḷiyaphagguna said to the Blessed One: “Venerable sir, who consumes the nutriment consciousness?”

3“Not a valid question,” the Blessed One replied. “I do not say, ‘One consumes.’ If I should say, ‘One consumes,’ in that case this would be a valid question: ‘Venerable sir, who consumes?’ But I do not speak thus. Since I do not speak thus, if one should ask me, ‘Venerable sir, for what is the nutriment consciousness [a condition]?’ this would be a valid question. To this the valid answer is: ‘The nutriment consciousness is a condition for the production of future renewed existence.

When that which has come into being exists, the six sense bases [come to be]; with the six sense bases as condition, contact.’”

-- SN 12.12 (Bodhi, trans)

and/or does this idea have any specific names/nouns (e.g. in Pali, Tibetan, and/or Chinese) associated with it?

The term for the stream of consciousness is viññāṇa-sota:

Again, having done this and gone further, he comes to know the unbroken stream of human consciousness as established both in this world and in the next. That is the third attainment. Again, having done this and gone still further, he comes to know the unbroken stream of human consciousness that is not established either in this world or in the next.

-- DN 28 (Bodhi, trans)

The term for the consciousness that links back to a new rebirth is paṭisandhi-viññāṇa.

The term for the process of the mind is citta-vīthi.