1) Lauzun's Legion was originally one (the 2nd) of the eight foreign legionary corps to be formed by the Ministry of Marine – the French Admiralty – for service overseas. Eventually, the project was abandoned after only three legions were formed: the 1st was assigned to the Caribbean; the 2nd to Africa; and the 3rd to the Indies.
2) Each legion was supposed to comprise a company each of hussars, gunners and workmen, supporting a small infantry battalion of two flank (grenadier and chasseur) and two fusilier companies. There was also a Compagnie Generale (a sort of HQ formation) comprising just one company of hussars; this unit is thought to be the one armed with lances.
3) The 2eme Legion became "Lauzun's Legion" around 1779/80 and included the Compagnie Generale. It was this formation that accompanied Rochambeau from France to Rhode Island – however, because of the lack of shipping, some of the infantry were left behind and only the two hussar companies (the CG and 2eme), the artillery company, and the two flank companies, were taken – and even then the hussars had to leave their horses behind.
4) At Yorktown, Lauzun's Legion had about 600 men – 300 were hussars, 200 were infantry, and about 90 were gunners.
5) A small group of the 1er Legion Etranger de la Marine came up from the Caribbean with St Simon. I believe it consisted of a group of about 50 hussars (without horses); they served on the York side of the river, not the Gloucester side, and so were not added to Lauzun's corps at any point.
6) Although these legions were composed of foreigners, they were predominantly German in character. It is often stated that they included many Irishmen; this is a myth, there was only one at Yorktown (an officer), whereas by contrast there were two Englishmen in the unit!
7) Here is a useful website for other info and uniforms:
lauzunslegion.com