If you have a connecting flight with a short stopover that leaves the EU within 24 hours after your departure, the Customs of the first departure airport will typically be able to validate your tax-free form.
If you are making an extended stopover enough to leave the airport or continuing your trip to another EU country before flying home, you will need to print the Airvat tax-free form in order to get a manual Customs stamp at your final EU departure airport.
For example, if you are flying from France to Italy and spending a couple of days there before flying back home to the USA, the Customs officer in Italy will have to manually stamp your paper tax-refund form before your departure for the USA.