Vichy France had control over most of the overseas empire of France, typically the fighting between Vichy France and Free France was negligible and the most significant military actions in the French overseas empire were fought by Vichy French troops against American and Commonwealth(British Empire) troops.
The reason for most of the overseas empire remaining loyal to Vichy France was because they were the legitimate government of France, and were led by many prominent French military leaders, such as Petain who was a famous French Field Marshal from WW1, and the President of the 3rd French Republic before it became Vichy France. Francois Darlan the head of the French Navy was the Prime Minister and second in command of Vichy France. In comparison De Gaulle the leader of the Free French was only a Brigadier General when the war started, and was briefly minister of Defence before the French Surrender in 1940.
French Somaliland was originally ruled by Deschamps with Brigadier General Legentihodme leading the soldiers stationed there, they had refused to accept the armistice, so Vichy France sent General Germain to restore Vichy rule in Somaliland, the local administrative council voted to remain loyal to Vichy France and Legentihodme fled to British controlled Aden. The British then blockaded French Somaliland, with the aim of convincing the garrison there to defect to the Free French, these attempts were almost completely unsuccessful until the allied capture of French North Africa in November 1942, when around a third of the native Somalis who were a part of the garrison defected. Then the Allies sent an ultimatum to Somaliland that they would invade, but if Somaliland surrendered it would remain a part of France after the war ended, which prompted the surrender of French Somaliland in December 1942.
In French Equatorial Africa most of the colonies defected to Free France in August 1940, with Gabon being the only colony that remained loyal to Vichy France, in Gabon the governor Georges Masson had agreed to join free france, however the population of Liberville opposed this and lead by the local Bishop Louis Tardy forced Masson to remain loyal to Vichy France, this prompted the troops in the Free French colonies of Chad, Congo, and Cameroon to invade Gabon, with the support of Commonwealth troops. After some mild fighting with less than fifty deaths Gabon was captured by Free France in November 1940.
The most significant battle for Africa between French forces was the Battle of Dakar, which was a naval battle fought over control of French west Africa in September 1940. Most of the allied ships in the battle were British, although a few Free French ships were involved as well. The Vichy fleet won the battle and maintained control over Dakar until November 1942 when Admiral Darlan who was the head of the French navy switched sides, and French West Africa defected along with him.
North Africa was the most important French colony and was invaded by the allies in the November of 1942 in Operation Torch. This invasion was assisted by the Geo Gras Group, a French Jewish resistance group that was operating in Algiers, and which took over key areas in Algiers before the invasion which led to the allies taking over Algiers without any significant fighting (although there was fighting in the rest of North Africa). Admiral Darlan happened to be in Algiers when the invasion was taking place and was captured, he reached an agreement with the Allies that he would join the Allies, in exchange for retaining control over French North and West Africa. Darlan was assassinated a few months later allegedly by a French monarchist.