Originally posted by Janus
There are quite a number of moon in the solar system which are being "pulled in " by their respective planets, the majority because they have retrograde orbits (They orbit in the opposite direction of the Planet's rotation.)
A lot of these are small captured asteroids.
There are a few examples of moons that actually match the pattern of having direct orbits with periods less than the rotation period of the planet, notably, Phobos of Mars, and Metis and Adrastea of Jupiter.
The Uranus system is odd in that it is the planet that rotates retrograde while the majority of the planets orbit directly.
As to whether a moon will break up and form a ring system once it passes within the Roche limit, this also depends on the size of the Moon. The Roche limit generally only applies to bodies that are large and massive enough for their own gravity to force them into a spherical shape. Bodies much smaller than this can pass within the Roche Limit without being torn apart. Examples of this are Metis and Phobos, both small, irregularly shaped moons that are actually already inside their respective Roche limits.
The majority of moons that are presentally approaching their planet are these small bodies.
Of the large bodies, we have Miranda, Ariel, Umbrial, Titania, and Oberon of Uranus, and Triton of Neptune.
These bodies will eventually pass inside the Roche limit, break up and contribute to these planet's already existing rings.