What are the rules for a non-EU family member travelling with an EU national?

Yes. Since you mention a type C visa, I assume that you are in a Schengen state. You can travel together with her to another Schengen state as long as she has her passport.

To travel to a non-Schengen state, she will generally be required to hold a visa for that state. Such a visa should be granted according to an accelerated procedure and free of charge.

There is one caveat, however. A non-EU mother in law falls under the freedom of movement directive only if she is dependent on the EU family member (that is, on her adult child or adult child's spouse). To get a freedom-of-movement visa for a non-Schengen EU state, she will have to submit proof of dependency (or of being a member of your household) as well as of the relationship. Similarly, if she is suspected of overstaying in the Schengen area, she might need to offer the same proof in her defense, although a verbal explanation of the circumstances would probably suffice in most cases.

In principle, however, her right to travel under the freedom of movement directive is independent of any document. This is why the document is called a residence card instead of a residence permit; it simply evidences a right that she already has rather than being the instrument through which that right is granted.

The relevant EU legislation is Directive 2004/38/EC.

Finally, if you are traveling out of the Schengen area, she will probably need to get a new Schengen visa to re-enter it. This would complicate matters somewhat, and, if I were you, I would postpone the trip if possible or at least discuss it with the relevant authorities in your country of residence before making definite plans.