The thing that worries me most is CITES, which controls the import of protected species, including ivory, abalone and many hardwoods which are often found in musical instruments. The EU is a single area so CITES doesn't apply to travel within it, so we were OK before, but now it applies. There is supposed to be an exemption for musical instruments, but making sense of the rules is a nightmare. It is possible to obtain a Musical Instrument Certificate which permits you to travel with instruments which contain protected species, but that seems far from easy (it asks for the scientific name - I know the fretboard of my guitar is rosewood, but how am I supposed to know what species?). I don't know if it's possible to get an MIC to confirm that your instrument doesn't contain protected species, in case you come up against a zealous customs official.

If you're touring professionally you'll need a carnet listing all your equipment and a work visa, with different countries setting their own rules for these. That's a whole different world of pain.