Vichy France was allowed to retain most of its colonies in the world ( Africa, Asia, South America, Middle-East and North American island of Saint-Pierre-et-Miquelon ). Which quickly became a problem for Vichy France actually as the loyalty of the colonial governments was not always assured.

To be more assured that the colonial government would be loyal to Vichy France and Germany, Pétain quickly replaced some members of the colonial administration such as in Tunisia or Algeria.

However, following De Gaulle's call in June 1940 some colonies immediately rallied to the new Free France. The first to do so were the territories in India, followed by various Pacific islands. The 26th of August, Tchad proclaimed its allegiance to de Gaulle followed by Cameroon. In French Congo, the colonial governor who was close to Pétain was arrested and replaced by General Edgard de Larminat one of the first soldier to join the Free French forces.

But many of the biggest colonies remained under Vichy's control such as North Africa, the small territories in China, the Middle-East, Indochina or Occidental-Africa.

Afterward, British and Free French forces landed in other colonies where they met more or less resistance from Vichy authority. St-Pierre-et-Miquelon did oppose resistance when Admiral Muselier landed. However in Gabon for example a 4 days long battle was waged between Vichy authorities and Free French forces/British troops.

The Syrian colonies saw ( if I am right ) the fiercest fightings. The Free French launched almost all they had alongside Commonwealth troops ( Australians and Indians for example ), Palestinian loyalists and even Free Tchecoslovaque troops against Vichy and colonial troops. It lasted for more than a month and resulted in some 3,000 allied deaths. The 14th of July though, Syria and Lebanon were under Free French and British control.

That's only a couple of examples. In general, many colonial were in support of Vichy. As we saw, some administration joined Free France and the allies without hesitation while others had to be pressured to do so or be taken by force. Many soldiers and officers of the Free French Forces came from the colonies and it became a valuable asset as they knew and understood the terrain which would be useful for the future battles in North Africa.