How does gradual crossing over of the Roche limit transform a planet or moon? Some cartoon simulations (an example) of an object (assume here a large moon with Earth-like density) which crosses the Roche limit of a much more massive giant planet, illustrate a circle suddenly disintegrating and forming a ring. But such a process occurs gradually over geological time scales. I doubt that there would be a single bad day when the whole moon disintegrates (as in -Oh, there Mt. Everest broke off and flew away!)

Wouldn't the tidal tugging cause volcanism and gradually melt the moon as its (somewhat eccentric) orbit spirals inwards across the limit during millions of years?

How would the melting moon deform? Would it really assume an oval shape, elongated towards the primary planet? Since the near side wants to orbit faster than the far side, wouldn't that cause the moon to spin at an accelerated rate, even if it initially was tidally locked? Would the melting, deforming and spinning save it from disintegrating further to within the Roche limit?

Some examples of Roche limit crossings I know of:


Comet Shoemaker-Levy 9 may have broken up suddenly because it had
such a high velocity relative to Jupiter.
Phobos will cross its Roche limit to Mars in ~50 million years. It
will hardly be a sudden process. I suppose its very low density and
mass won't allow any volcanism and melting, though.
KOI1843.03, an exoplanet candidate with a density of ~7 g/cm³,
is already inside the Roche limits for its less dense compounds which
it has shed (as I interpret it).
A large moon (the size of a planet) would be modeled by the fluid model, meaning friction and tensile strength are too weak to modify the shape of the moon significantly, shape is determined by rotation, vortices, self-gravitation, and tidal forces.
Smaller moons, like Phobos would likely follow a rubble pile model.
On circular orbits tidal heating doesn't play a major role.

The fluid model has been studied extensively. The occuring shapes are known as equilibrium figures.
For a full understanding, the parent body should be considered together with the moons, as one system. (Fluid) moons getting too close to the Roche limit gradually deform (tidal heating depends on inner friction), and start to shed mass at some point. There exist solutions with vortices, and vortex-free solutions, depending on the properties of the fluid. The lost mass may either form an n-body system outside the Roche limit, or form a ring system, mostly inside the Roche limit.
Closer to the planet, it may either impact, and fuse with the planet to a the overall spheroid of the planet, or - if the planet is already rotating fast, and the mass of the moon was sufficiently high - increase the rotation of the planet, such that it may undergo e.g. the Maclaurin - Jacobi series, up to the point, that it cannot accrete more mass. There are several more possibilities in detail.

Rubble piles with tensile strength are more likely to undergo sudden shape changes or disruptions than rubble piles without tensile strength. Behaviour of rubble piles without tensile strength depends on friction. The less friction, the closer to the fluid model.
The more tensile strength of the rubble pile, the more it behaves like the ridgid model. Ridgid objects may survive well below the Roche limit for fluid objects, but disrupt suddenly when tidal forces get too strong.