when can police tell you "not to leave town?"

As a suspect, prime witness, or as a defendant released on own recognizance, or on bail--one isn't supposed to leave town. Steps to insure this could be a tracking bracelet. Other possible 'can't leave town' ideas: dying beloved relative, MC needs dialysis or a rare drug infusion from a certain clinic.

I have no personal, or professional, experience to impart, but I always thought that was just some dramatic tag line. I assumed (ass*u*me, I know) that they said that because it would look suspicious, but not because they had any legal authority to remand you. Unless you have been charged, aren't you still free to do whatever the heck you want?

Now, if they questioned her, your character might feel compelled to stick around so as NOT to look guilty, but still, no legal ramifications otherwise.

I don't know. Hopefully someone with more knowledge on the matter will chime in.

As long as there isn't a bail issue, it's more of a "leave town, and it makes you more suspicious" thing. That is, it has no legal weight, but can lead to legal issues (especially if it's found that the person was hiding evidence or got out of town specifically to hinder the investigation).

If it helps...
FR

Unless there is a court order on your location, you are free to leave. It is an old TV show thing to have the po-po say 'don't leave town' tho it did used to happen 'til someone finally cracked a book and noticed there was no legal requirement to stay in town if you're a suspect. A theft suspect who realizes he's 'bout to get caught would be a smart boy to leave town. If a cop had told him not to skedaddle, that cop too would have watched too many old TV shows. While there really is nothing to prevent the cop from saying not to leave, there's nothing to make it stick. The suspect is perfectly free to skedaddle. A charged person on the other hand, may be released OR (on recognizance) or to a third party with mobility restrictions, but that is to ensure he makes court appearance. He still does not have to cooperate in the police investigation, tho at that point the cops had enough to charge, so the investigation is largely done.
Pretty please, do not have your guy stay in town 'cause copps said he couldn't leave. They cannot slap a ankle bracelet on a suspect, they cannot keep him from giving the one-fingered salute and catching the next plane--unless they are willing to arrest him. The only way to stop him is to charge him, go ahead and effect the arrest.