Right. They throw away their documents, so EU countries have to document and fingerprint them again and also keep a database, preferably a shared one. Otherwise they just conveniently move from one country to another as there are close to no border checks in the Schengen area. Probably why Germany has reintroduced "temporary" border checks after the Solingen stabbing attack. Sweeden still has checks in place at the bridge to Denmark.
My acquaintance managed to sneak in multiple times, “losing” their passport if caught, then pretending not to understand fingerprinting instructions and smudging them on purpose until the police gave up. With face biometrics probably won’t work today. Just saying that a stamp in a passport is necessary a full proof solution.