I came across this problem the other day when being asked to conduct base training in a business jet to a newly qualified pilot in an EASA member state. Since the ATO, where the pilot had done his initial course, couldn't help with doing this, his employer came to me.
Their first question was whether I was affiliated with any ATO (which I was not). Then they contacted their civil aviation authority and - voilá - they granted me a one-time permission to conduct this base training.
Maybe other CAAs will start dealing with these 'challenges' in a similar way...