How can I calculate how the debris of an object ripped apart at the Roche limit will spread out? Let's say I have a moon orbiting a planet at a distance $d$. Eventually, $d \leq d_R$, where $d_R$ is the Roche limit of the moon-planet system. I can figure out the mass of both bodies by experiment, as well as their radii, and thus their density (which is how I can calculate the Roche limit).

I'm curious as to how the debris will spread out, though, once the moon is ripped apart. Can I calculate the inner and outer radii of the newly formed ring, based on what I know? Or is there not a direct relationship between these radii and the initial conditions?
I think there are two key aspects to the answer.  1) Solid/rocky bodies should tend to collide before they reach the Roche limit.  2) When gaseous bodies reach the Roche limit (and undergo 'Roche-Lobe Overflow'), the dynamics are basically those of test-bodies and are fairly straightforward and well understood from binary stellar dynamics.  To expand on both:

1) Rocky Roche Limit.  If you are thinking about a gaseous donor (the object being disrupted), then this is an irrelevant point, but It sounds like rocky is what you had in mind.  To an order of magnitude, the Roche limit is the same as the Hill sphere, or the Tidal Radius (e.g. Rees 1988) --- which is simply the radius at which the density of the donor equals the average density of the primary in a sphere of that radius:

$$R_t^3/M \approx r^3/m \rightarrow R_t \approx r \left( M/m \right)^{1/3}$$

Rocky material varies very little in density (e.g. Iron meteorites are only about twice as dense as chrondritic ones), which means that the densities will only match very near the radius of the primary.  The tidal bulge in the secondary (donor) also makes it easier for the objects to collide before meeting this criteria.

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2) Gaseous Roche Lobe Overflow.  Basically every text on stars and star systems will have a section on mass transferring binaries which will describe the dynamics of mass transfer (this PDF is by Philipp Podsiadlowski who is a wizard of the field).  In the type of situation you are describing, like in (semi-)stable binary systems, it is a very gradual process where material is slowly syphoned off of the donor.  This material can either form an accretion disk (high angular momentum material) and gradually accrete onto the primary, or directly impact the primary (low angular momentum material).

e.g.