How does rebirth work for arahant and non-arahants?

"Re-birth" is the re-arising of self-view after the traumatic "death" of a previous self-identity. For arahants, there is no birth & no death therefore the question about what happens after "death" ("marana") is irrelevant. But for non-arahants, "death" ("marana") still occurs. When self-view is still capable of arising, then a new birth will happen after death. Whenever attachment arises, there will be both birth & death, as follows:

Seeing danger in clinging, in the coming-into-play of birth & death, they are released from lack of clinging, in the ending of birth & death. They, happy, arriving at safety, fully unbound in the here-&-now,

MN 130

In summary, the word "marana" ("death") does not refer to something "physical", as shown in the quotes below:

21. Heedfulness is the path to the Deathless. Heedlessness is the path to death. The heedful die not. The heedless are as if dead already

Dhammapada

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Bhikkhu, ‘I am’ is a conceiving; ‘I am this’ is a conceiving; ‘I shall be’ is a conceiving; ‘I shall not be’ is a conceiving; ‘I shall be possessed of form’ is a conceiving; ‘I shall be formless’ is a conceiving; ‘I shall be percipient’ is a conceiving; ‘I shall be non-percipient’ is a conceiving; ‘I shall be neither-percipient-nor-non-percipient’ is a conceiving. Conceiving is a disease, conceiving is a tumour, conceiving is a dart. By overcoming all conceivings, bhikkhu, one is called a sage at peace. And the sage at peace is not born, does not age, does not die; he is not shaken and does not yearn. For there is nothing present in him by which he might be born. Not being born, how could he age? Not ageing, how could he die? Not dying, how could he be shaken? Not being shaken, why should he yearn?

MN 140

SN 12.66 say unambiguously the cause of "death" is "attachment" or "acquisition".

As he explores he understands thus: ‘The many diverse kinds of suffering that arise in the world headed by aging-and-death: this suffering has acquisition as its source, acquisition as its origin; it is born and produced from acquisition

SN 12.66

In Buddhism, "a being" ("satta") is merely an "idea" or "view", as follows:

Why now do you assume 'a being'? Mara, have you grasped a view? This is a heap of sheer constructions: Here no being is found.

Just as, with an assemblage of parts, The word 'chariot' is used, So, when the aggregates are present, There's the convention 'a being.'

SN 5.10

"Death" ("marana") refers to the death of "a being", which is merely an idea or "self-view", as follows:

And what, bhikkhus, is aging-and-death? The aging of the various beings in the various orders of beings, their growing old, brokenness of teeth, greyness of hair, wrinkling of skin, decline of vitality, degeneration of the faculties: this is called aging. The passing away of the various beings from the various orders of beings, their perishing, breakup, disappearance, mortality, death, completion of time, the breakup of the aggregates, the laying down of the carcass: this is called death. Thus this aging and this death are together called aging-and-death.

SN 12.2

Yes, there is ultimately no self in all phenomena, so there is no self that is reborn.

However, there is suffering. And suffering can only be ended by Nibbana. Suffering doesn't end with physical death. Suicide cannot end suffering.

But if anatta is valid not just for arahants, but for all conditioned phenomena as well, does that mean that rebirth is also not applicable for non-arahants either?

Just an analogy, a novice martial arts student at the beginning was told to practice hard and adhere to all the strictest forms and moves in order to make progress on the path to become the best fighter. But once s/he's reached master level, then s/he'd be told to be fluid, adaptive, and NOT to adhere to the strictest forms and moves to become the best fighter! They're not contradicting as they sound. Similarly, we unenlightened worldlings have not been able to directly experience the non-self truth for ourselves. The arahants do. And that's why we worldlings still need to adhere to those strict "drills" to get to the next levels. Until we've directly experienced the level of "no one to be reborn", then we'll earned the right to proclaim that rebirth no longer applies to us.

Indeed puthujjanas knows that it makes no sense to ask where a fire go once it is put out, unless those puthujjana managed to create a fantasy where they deify fire..., so the buddha says that It makes zero senses to ask where an arhant goes, or if an arhant exists after parinibbana, by creating a parallel between fire and craving (or fire and existence). Same thing about the stupidity of trying to figure the non-existence of a buddha.

It is craving that is the path to vedana, the 5 aggregates, birth, dukkha and all that stuff, so once there is no craving, well there is nothing left and there cannot be birth again.