> Does this actually happen?
Absolutely it does. If you live near the Mexican border of the US, it’s extremely common to see older cars being towed headed south, presumably to Mexico. It’s not particularly expensive to drive them down to the border.
> I bet the cars get trashed for scrap or parts.
Only the least valuable and oldest ones that other countries don’t work. That means you’re overall moving up the modernness of the fleet. Presumably with better fuel economy, lower emissions, and fewer problems.