Under the settlement between Vichy and Germany, the French colonies were considered as part of the unoccupied 'Free' zone of France. People living in them would have enjoyed far more freedom of movement and expression than someone living in Paris or Normandy would have. However, the colonies were still subject to the reactionary Vichy regime and political freedom was still tenuous. For example, when a number of French legislators left France to go to Algeria, Petain had them arrested upon arrival so that they could not set up a rival government. Also, given the disproportionate power in the relationship between France and Germany, Germany was often able to impress its will upon France in regards to the colonies. Partly, it was Germany's expectation that the colonies could and would be used to support the German war effort. Vichy France on the other hand felt that maintaining the independence of its colonies was an important expression of sovereignty. Additionally, DeGaulle was very focused on incorporating the French colonies into the political structure of the Free French Forces. These conflicting interests eventually led to a complicated life on the ground. For example, Germany was able to force Vichy to allow them to use airfields and transportation infrastructure to support their forces in North Africa. Germany also used Syrian airfields to support an Iraqi revolt against the British. After this, Britain invaded and occupied French Syria. Eventually, when the Allies landed on the North African coast Admiral Darlan, who was the head of the French colonial military in Algeria, was convinced to switch sides and surrender the colony. This precipitated a total loss of confidence in Vichy by the Germans, and they subsequently occupied all of mainland France.

In summary, the Nazis were content with delegating the administration of the colonies to Vichy. Vichy regarded this as an essential concession because they felt that the colonies were an important expression of French power and national pride. However, the colonies were also courted by the Free French, and would become embroiled in the broader fighting of the war. Nonetheless, the French colonies which were administered by Vichy were left mostly autonomous of Nazi control. The movie Casablanca portrays some of this complexity, as it would not be unreasonable for a Frenchman who could not live comfortably in Paris or occupied France to immigrate to Algeria or Syria.

/u/nate077 is on point regarding North Africa and Syria, but the situation in the Sub-Saharan African colonies was a litte different.

France had organised its colonies in Sub-Saharan Africa in two large federations: French West Africa (Mali, Mauritania, Senegal, Guinea, Burkina Faso, Togo, Benin, Ivory Coast, Niger) and French Equatorial Africa (Chad, Central African Republic, Congo, Gabon, Cameroon).

These colonies were federated but operated semi-autonomously. Each territory, the future countries, had a governor, answering to a Governor-General.

In French Equatorial Africa, the most influential figure is arguably Félix Eboué. Eboué was born in French Guyana, in South America, the descendant of African slaves, and had a promising early career as an administrator in the French Caribbean and in French Africa. He was appointed governor of Chad in 1938, with a mandate to fortify Chad as it was a strategic highway towards French Congo.

In 1940, after Metropolitan France fell, Eboué officially declared Chad as the first territory to follow De Gaulle. Chad didn't have much of an economy yet, but it was a huge symbol. Free France had no territories at that time, and De Gaulle was nothing more than a guerrilla leader propped up by a foreign government. With a territory behind him, Free France started to look more like a government-in-exile. The French military contingent in Chad followed Eboué, and as such, every soldier and officer in that garrison (including my own grandfather) were sentenced to death in absentia by the Vichy regime for treason. They would be pardoned after the war, of course.

Once Chad declared, every colony in Equatorial Africa followed suit, except Gabon which stayed in the Vichy camp, prompting the "battle of Gabon" or "battle of Libreville" in November 1940, lasting 4 days and leading to the capture of the Libreville garrison. Félix Eboué was named Governor-General of Equatorial Africa by De Gaulle, and raised troops and helped set up a base from which the Allies could launch attacks into Libya and Algeria.

In French West Africa, the Governor-General, Pierre Boisson, was a Vichy sympathiser, and his garrison fought the Allies in the "battle of Dakar", a naval engagement and an Allied defeat in september 1940.

An interesting anecdote about the Battle of Dakar should give you insight about the mindset of the time. The Allied ultimatum to the Vichy forces in Dakar accused them of planning to surrender their equipment to the Germans. It was met with indignation: In the commander's eyes, Vichy France was not Germany. It may have been a rump state, but it was independent. Pierre Boisson is quoted as saying "France gave me Dakar, I will defend Dakar."

Anyway, West Africa stayed in Vichy's camp until Algeria fell to the Free French Forces. At this point, surrounded by enemies, Boisson surrendered and West Africa declared for the Allies.

That's basically the end of French African territories' involvement in the war - they were a base of operations, repair and manufacturing for the rest of the war, but didn't switch sides anymore and weren't fought over.

However, it should be noted that large contigents of "indigenous troops", Arabs and Africans, were raised and fought in some of the most brutal engagements of the last years of the war, but that's another topic.

Source: Pierre Montagnon, "La France Coloniale" (1990)