Just my opinion, view and experience...
Although recently retired as a full time Bassist, I had previously worked on dozens of resident shows, theatre tours and concerts with touring acts in the EU since the mid 70's (some at a minutes notice). There were no visas or ATA Carnet forms needed, or similar issues of that kind. Sometimes, just turning up at airports or Waterloo station with a suitcase and Bass to collect tickets. It was as easy as arriving for shows/tours and gigs in the UK. But unfortunately, the simplicity of working as a UK Musician in the EU, looks like it is coming to an end.
Before that, it wasn't as plain sailing as some make out. It was littered with Visa applications (some a pain in the derrière), Carnet documents and procedure, bonds and the occasional sponsorship (different Countries had different criteria and requirements). Although, it was usually organised by UK based fixers and agents in advance (characters like Jack Fallon of Cana Variety Agency - Ironically, a Canadian. There were plenty of others, names escape me at the moment).
But still, it didn't stop you getting hold ups, grief, cases/gear being opened up and then poked about at various European borders. It was usually, not a pleasant experience and always defiantly, time-consuming. Buying, selling, or replacing equipment was always a hassle with customs and the Carnet forms. A lot of the younger musicians in this field are going to miss the benefits and ease of Freedom of Movement across (EU) Europe. Maybe even resulting in levels of work dropping for UK Musicians wanting to work over there, due to the extra work and cost involved.
Howard Goodall wrote an interesting piece about the possible fall out...
http://www.howardgoodall.co.uk/articles-press-etc/the-musicians-passport