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Not as many problems as traveling without one.
Having traveled before with someone in that exact situation, yes, there will be harassment and TSA may very well intentionally force you miss your flight and end up waiting to fly standby on the next one. But as of now, you generally do get through, which cannot be said for traveling internationally without any passport at all.
Will that change? Highly likely. Soon? Probably.
But there are a not insignificant number of people who only recently decided it’s no longer safe, applied for passports, made plans to leave, and saw recent stories about passports not being issued. This type of scare mongering does not help them, and can be actively harmful to their mental health and well-being.
Edit: Obviously the longer they stay, the higher the chance this changes.
See my comment to the other reply here