I have a multiple-entry French Schengen visa (valid for almost 5 months, allowing 90 days of stay), which I have used twice to enter France, using up a total of 10 days. I am now considering traveling to Belgium or the Netherlands.
If traveling by Eurostar, would I present my passport to the border police of France, or would it be the authorities of the destination country, say Belgium or the Netherlands? I am asking this question because it seems that there are Eurostar services that do not call anywhere in France - they proceed directly across the border to Brussels.
Irrespective of the first question, am I allowed to travel to other Schengen countries, possibly by air, after using my French Schengen visa twice to enter France?