Thankfully never lost my passport but have at times travelled with one of my other two as backup "just in case" - similarly the Mrs has to (for all intents and purposes) carry both of hers when going home to see the folkes as is required to enter her home country on her home passport and using her British passport is by far the easiest way to get into the UK (the passports are in different names so also helps having it as proof of ID as it matches all her bank cards etc).
No idea if there'd be issues of using different passport than that provided for the advanced passenger information or border issues for countries that have more formal exit procedures.