When a professor goes on unpaid leave from her university, can she normally continue to use her grants to fund graduate students and postdocs? Or, do all grants "freeze" until the professor returns?
I don't see any reason that a professor wouldn't be able to continue to use grants while on leave. In fact, I'm on unpaid leave from my home institution right now, and have been able to use my grants (for travel support for myself), so that's one data point.
The answer is "yes". For instance, I was on unpaid leave this academic year, and used my grant to support a graduate student.
It depends on the breakdown of the grant. If there are students or staff who were hired to do the proposed work then I don't see any reason why they wouldn't continue being paid if they are continuing to work. The same goes for research related services and products. Sometimes sabbaticals are unpaid, but that doesn't mean that the work stops. Most grants are on a schedule, so PIs need to be making some form of progress on the project (whether or not they are getting paid from it).