I can only speak to the question of battles, there were no major battles that occurred but there were some conflicts. There were battles in Canada (Quebec, Nova Scotia) fought by the colonial armies. Almost everything outside of North America was naval related. There was a raid on the port of Nassau in the Bahamas. Captain John Paul Jones commanding a single ship defeated a single British ship in the North Channel in 1778. The next year and aided with several French ships, he achieved perhaps the greatest victory in Great Britain, again in the North Channel.

After France joined the war with the 1778 Treaty of Alliance and Spain allied itself with France in 1779, there were a more battles between them and British forces outside of North America. The Battle of Ushant in 1778 centered around islands in the English Channel. The Siege of Pondicherry occurred in India with the French surrendering after about 10 weeks. There were five other French-British naval conflicts in the early 1780's that occurred near India. French and Spanish forces unsuccessfully tried to capture Gibraltar from the British from 1779-1783. There were numerous smaller incidents in the Caribbean between French, Spanish, and British forces.