Another name for the flintlock musket was the fusil. Fusil is a French word that means, um, flintlock musket.
A soldier who used a fusil was called a fusilier.
[Pictured: the uniform of the Royal Welsh Fusiliers in the 18th century, American Revolution reinactment. Notice they are carrying Land Pattern ("Brown Bess") flintlock muskets of the period.] Click picture to enlarge.
Flintlocks were used since at least the English Civil War (1642-1652.) Early military organizations of foot soldiers were simply called infantry. One early (1712) unit to be called fusiliers was the 23rd Regiment of Foot - later called the 23rd Royal Welsh Fusiliers. The Royal Welsh Fusiliers served in the American Revolution.