Like so much else surrounding Brexit, the answer to this question is unknown. However, unless and until an agreement to the contrary comes into force, on the date of the UK actually leaving the EU (in around two years' time), strictly, a UK passport holder will not have a right to stay in an EU country longer than any visa strictures allow (if a formal visa...
When travelling between France and UK (did this so very many times) I always had to produce my passport. Going to UK I had it checked in France and had to check it through at Leeds/Bradford airport. Coming back I had it checked in UK and also, as I queued relatively patiently, at Limoges airport, in France. What's different?
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