The West Indian islands often changed hands, as they had some agricultural riches but were poorly defended in the seventeenth and eighteenth centuries.

The consolidation of the european colonies in North America did not only involve the Dutch in New Amsterdam. The British also took New France (Quebec and a lot of territory sparsely populated by europeans, in what is now eastern Canada and the north east and mid west of the United States) from the French and a Swedish colony somewhere around New Jersey/Pennsylvania. There were probably others as well.

As to the Dutch idea of freedom. Well maybe in New York. However the United Provinces were a collection of pretty conservative cities and aristocratic republics. The English ideas of elected assemblies seeking redress of grievances from the King and traditional liberties were probably more influential in the other twelve colonies with little or no Dutch influence.