NATO is concerned about flooding in Finland. It researched flooding in the Gulf of Finland. NATO held a flood response exercise in Finland. Oh, and it's also concerned with Ukrainian, Moldovan Bulgarian, and Slovakian flooding too. And I would be remiss in ignoring NATO's contributions to helping Kyrgyzstan cope with snow.
These are perfect examples of some nice, cuddly jobs that an EU-centric military alliance would be good at in the near-term, and probably even more so after the dissolution of NATO when the Europeans will be forced to buy and train real militaries.
Somebody asked me back when I was on my last kick how I picture a EU military organization working. I assume they didn't want to know about the inevitably huge bureaucracy producing documents in twenty languages, but rather how it would fight.
The Brits and French would probably make up the real bulk of the fighting forces. Despite the portrayal of the French as surrender monkeys, their forces, if properly equipped, could and would fight. The Germans would contribute some fighting forces, but I think their comparative advantage is with logistics. Any country that can keep their trains that well run should deal with logistics and material. Poland and other larger countries would provide more general forces, but I picture most of the smaller European countries developing expertise in a niche or two. Norway might be the alpine combat experts. The Czech Republic might field top-notch HAZMAT teams. Finland could deal with military communications.
This way each country has either a general fighting force or some specialty. If their specialty is needed for a certain deployment, then there will be political and peer pressure to commit forces to the effort. The biggest issues I see is the UK and France deciding on who would send their troops. They'd probably end up splitting the duties. That's assuming the UK joins into an integrated EU force.