I'm not sure if this is a red flag but it appears to me it is, as I have seen it occur more frequently by self-interested employers:

When you don't respond to their inquiry and they keep prodding you about it, even weeks after it.

That's just recruiters being recruiters usually. I often have a cut and paste "sod off" style response I put in there just to stop them hounding me with messages.

@AlmondSauce Haha.. it'd be interesting to see what kind of content those messages would have. An effective 'sod off' is good.

Though, my professor always said: "Don't burn your bridges". Granted, they aren't bridges.. they're just noise, paper weight.