What Europe does not have is any cross-national agency with the power to carry out its own investigation and make its own arrests.
This means that cross-border policing in the European Union has big holes. It depends heavily on informal cooperation rather than formal institutions with independent authority. Sometimes this works reasonably well. Sometimes this works particularly badly. Belgium is a notorious problem case, because its policing arrangements are heavily localized. In the past, many Belgian policing forces have had difficulty cooperating with each other, let alone with other European forces.
That is from Henry Farrell, there are other points at the link. Here is one bit more:
To take a different example, immigration and refugees present an even bigger and more visible set of challenges to the E.U. than terrorism, yet the E.U. has been unable to agree on reforms that might expand the budget and powers of FRONTEX, the E.U. agency charged with coordinating border control. Creating a European FBI-style institution would be an even bigger lift.
Keep this in mind the next time you are tempted to believe that the EU is doing everything possible to manage the refugees crisis well.