It is sure and demontrated that the French Dragons (more than all the rest of Europe) used to fight on foot, above all when it was matter of siege; these units - along with grenadiers - were considered the most valuable for trying to enter a fortress, due to their capacity also to fight in a sort of skirmishing and their professionalism.
It is also sure they fought on foot with bayonets and "as infatry", this is in deep ranks formations. Now the question is whether they fought on 3 or 4 ranks like the French Infantry. This questions arises because the numbers of a Dragons' squadron were quite limited, (around 100 men at full) and so their extended front (on 4 ranks) would have been quite narrow.
Here an example from the Languedoc I Dragon Regiment.
This unit at the battle of Turin fought on 3 squadrons:
1 Colonel
1 Lieutenent Colonel
2 Aides de Camps
3 Commanders of Squadrons
3 Oboists
10 Drummers
3 Guidons
4 Cornets
10 Captains
9 Lieutenants
28 NCO
203 Dragons