In my country (France) we have to be registered before changing € to $ USD with names, address etc... Is a US bill traceable? Can the authorities know with the serial number were the bill went, in which countries, and the name of the person who took the bill? Is the bill registered at a foreign country's border? I'm in foreign law college in France; we talked about money but not this kind of stuff and my teacher can't help me.
Bills can be traced and limited information about their history may be available if you have a bill--for example, you may discover when it was printed. In addition, some money counting machines will scan the serial numbers of money that is counted, and cash used for a ransom, for example, may have its serial numbers tracked. However, serial numbers are generally not tracked if a consumer simply takes a bill across a border. The authorities would need to scan every dollar bill, and that would be annoying and largely a waste of their time.