> Strangely enough, this is exactly the opposite of how it works. The dollars abroad tend to stay abroad, as either a more stable alternative to local currencies, or a reserve currency.

I think you have it backwards. The US dollar can be handled as a more stable alternative only if it's actually stable. As soon as the US government starts to deteriorate goodwill and outright be hostile towards the world, not only does the US dolar lose its value as a stable alternative but foreign governments start to be motivated to dump their assets, which further tanks its value. In the past couple of weeks we started seeing countries outright dump their investments in US dolar to derisk their portfolio, and they did it at the tune of billions of dollars. You also started to see political pressure for foreign governments to demand their gold reserves are pulled out of the US, which means this pressure goes way beyond US dollars.