It seems like millions of Spaniards or Portuguese emigrated to their colonies in the Americas, during and after colonization. Similarly, British America was much more populated than New France, and even after that a lot of people from Britain and Ireland emigrated there. There are also a lot of heavily populated former British colonies (Canada, Australia, etc.). On the other hand, lots of Italians, Germans, Jews emigrated to North America, South America or even Australia.
In none of those cases (be it in colonies, or after that in the republics of the New World) there seems to have been a large French emigration.
The only exception I can think of is Algeria, where there was some 1.3million pieds noirs, but that is not a lot (and a substantial share of them were of Spanish or Italian descent anyway).
Why did the French not emigrate in large numbers to their colonies or the New World, like other European countries did?
Historian Allan Greer, in his book "The People of New France", notes how the relative lack of immigration to "Canada" (now known as Quebec) seems odd. In the time between the first settlement in the New World and the British conquest of Canada, approximately 27,000 people came to Canada - and the majority of those returned to the metropole - only about 10,000 settled permanently (a "surprisingly small" number compared to the million who settled in the modern-day US - mostly from a country a quarter of the size of France).
Greer attributes this to a few factors:
(1) Canada had a very bad reputation in France. It was seen as an icy hellhole constantly threatened by Iroquois attacks. While exaggerations, conditions were not great in Canada especially for the "engagés" (essentially indentured servants).
(2) Conditions in France were relatively better than in many other European countries - peasants had a relatively more secure hold on their land. Additionally, the tradition was to divide inheritance equally amongst all the heirs; there was therefore not the proverbial second son who would have no anchor to the family home and would want to seek a better life in the New World.
(3) The French government banned the emigration of Huguenots (i.e. French protestants) to Canada. Many emigrants from England were religious minorities - but the French government shut off this potential spigot.