I'd guess that the topic only marginally differs in different geographical regions. I narrowed the question to government agencies, because privately funded projects usually have contractual obligations that may vary substantially on individual basis and that may also not be publicly visible.
Let's assume a university employed researcher (e.g. tenure-track faculty) successfully applied for a grant, i.e. an amount of money for a defined period of time intended for research on the topic stated in the grant proposal.
As time goes by, the work progresses, papers are published, students work on their theses, etc. There is no apparent evidence of any foul play. However, the PI in question embezzles a relatively small but significant amount of money for various purposes (own pockets, financing another project, lab equipment that is not related to the project, ...).
Under what circumstances would the funding agency suspect that there wasn't enough progress made / work done or a misappropriation happened with the granted funds and start investigating? What are the mechanisms for detecting such cases?
Note that I causally link lack of progress and misappropriation mainly for illustrative purposes and I'm not suggesting that one can't happen without the other or that embezzlement is the only reason for poor results (or the only illegal action for that matter). Indeed, the discussion regarding the opposite case, i.e. when embezzlement happened, but the project yielded excellent results nonetheless, would be interesting for another question. My question is primarily concerned with the when and whether a funded project can be deemed insufficient and thus lead to an investigation.