Auttie,
There were two types of dis-mounted dragoons: the first would involve the mounted troopers simply, well, dismounting in order to – for example – clear a building; the second – which I think is the type you are taling about – is something different.
In late 1775/early 1776, there was an "augmentation" which added about 20 NCOs and men to an infantry company, and a junior officer (cornet), a sergeant, two corporals, and 29 privates to each of the six troops in a Light Dragoon corps. (At the same time, another 9 privates were added to the mounted component of each troop.)
The men were never mounted and were intended to serve as light infantry from the start. As such, they were taught light infantry drill and armed with rifled carbines which could take a bayonet. The dismounted elements of each troop generally served together as a dismounted division, thus each of the two British light dragoon regiments was a legionary corps with a roughly equal mix of horse and foot.
Each of the two regiments in America, at full war establishment, numbered just under 500 men, including officers. The dismounted divisions, plus reinforcements for the mounted division, landed in America in September 1776, and both the 16th and 17th took part in the New York campaign. Some dismounted men of the 16th were at Princeton, and the whole of the 16th took part in the Philadelphia campaign and, later on, the battle of Monmouth.
When the 16th was sent home in late 1778, the dismounted men were transferred to the 1st Light Infantry; the senior sergeant actually became the Sergeant-Major (nowadays the Regimental Sergeant Major) of that battalion. Here's Don Troiani's interpretation of them as they would have looked on arrival in America:-
link
(You need to click on the title in the contents page, fifth one down.)
Does that answer your questions? If not, don't hesitate to ask more.