When a US registered airplane is exported to another country for registration in that country's CAA system, the aircraft must undergo a "conformity" inspection for the export authorization from the US state department (who has authority for exportation of certain, controlled, US mfr'd goods, including aircraft.)
The conformity inspection must be certified by either FAA (aviation authority of the exporting country) or their designee, such as a DAR. Only THEN may it be exported. (This is why some operators choose to keep their airplanes operated in foreign countries on the US registry. I recall some airplanes I've exported that had equipment not meeting the airworthiness requirements of the importing country, (i.e. unapproved equipment, mods, or installations, or sometimes simply missing documentation).
The consideration in that case is that only a US certificated pilot may operate a US registered airplane. (Explains why some foreigners have US certificates....they actually never operate over here, but they operate US registered airplanes "over there".)
(The reverse also must occur when re-importing a foreign registered aircraft to the US...even tho' it may have originally been mfr'd here. (My own N146YS went thru this excersize, and an obvious error was overlooked in the process...it had the wrong engine installed when it received it's first US airworthiness certificate. The FAA inspector misssed that detail.)

I discovered it when I bought the airplane and subsequently "legitimatized" the installation by purchasing Ron Massicot's STC before he donated it to our Assoc'n.)
It's also the reason US/N-registered airplanes in foreign countries must undergo their inspections by US certificated repairmen/APs.
On a related matter, .... I sometimes hear potential buyers become very nervous when purchasing a foreign-registered US-mfrd aircraft for re-importation. They are usually worried about shoddy maintenance.
But my experience has been quite the opposite, expecially if the aircraft has been on a British-Commonwealth country registration. Their regs are so much more insistent on factory-authorized parts/service that usually those aircraft are in much better condition than similar US registered aircraft. In fact, if a US mfr'd aircraft is kept on the US registry while operating in the foreign country.... it's usually in order to avoid the more stringent maintenance issues of the foreign CAA. (A Pt 91 US registered airplane has considerable more options for mx practices than most foreign CARs, which commonly insist on the same rules for both private and commercially-operated aircraft.)