How does cutting work: Does the laser change it's focus as it cuts deeper and deeper?

I am assuming that the beam is focused to a point right at the surface of the material. As the surface evaporates, the beam must be focused slightly lower to evaporate the next layers, and so on until the cut goes all the way through the material. Is that correct?

Or does the beam get focused into a narrow beam, not a point, that can cut all the way through anything in its path regardless of height and thickness, and the beam leaving the bottom of the material is still condensed and able to cut, but the tray prevents that?

The beam does not focus to an infinitely small point. It is cone shaped as it leaves the lens but it has a small waist like an hour glass that is perhaps 0.1" long and 0.008" diameter and then it spreads out again tending to a cone. It is generally focused on the surface but the depth of the cut is limited by the fact it gets out of focus the lower it goes.

You can do another pass at a lower focus height but then the cone is clipped by the top of the cut, which is what stops you cutting an arbitrary depth.

Some materials have internal reflection that allows it to cut deeper than it would otherwise.