Dragoons frequently fought dismounted, and, in fact, at Naseby in the first civil war in England, dismounting the dragoons and putting them at an oblique to the parliamentary foot, effectively flanking any advance by the royalist foot, proved crucial in the New Model's victory. Until quite recently in modern times, other cavalry operated in much the same manner as dragoons did when they were mounted. Cromwell's cavalry, as did all cavalry in the 17th century, would approach the enemy at a walk, and then fire their pistols (horse pistols were huge things, only slightly smaller than the carbines adopted by dragoons in the 19th century). With notable exceptions such as Cromwell's cavalry, a cavalry charge with swords drawn (both heavy cavalry and dragoons used a military long sword rather than a saber--sabers didn't become common until the 18th century) inevitably meant that that body of horse was useless for hours, and usually the rest of the battle. At Marston Moor, Rupert's cavalry charged, broke the parliamentary foot, and then carried on in a mounted, disorganized mob to loot the parliamentary trains (which they considered their legitimate prey). By contrast, Cromwell's horse charged, and then rallied, reformed, and charged again. Such discipline by mounted troops did not become common until the reorganization of the French military in the late 18th century.
Dragoons were most commonly used in
la petite guerre, the patrolling and raiding operations which went on constantly as armies marched and countermarched seeking an advantage. Joining the dragooms was popular because of the opportunities for looting. Dragoons were considered ideal for the work because they could dismount and confront infantry formations on an equal footing if encountered during their patrols.
Dragoons operated well into the 19th century, in both capacities--as vedettes to patrol the area of campaign, and as cavalry on the battlefield. At the first battle of Balaclava, General Scarlett's Heavy Brigade made a crucial charge on the Russian horse. His brigade included at least three dragoon regiments, the Royal Dragoons, the Fifth Dragoon Guards and the Sixth Inniskilling Dragoons. The Scots Greys (the oldest continually serving cavalry regiment in British service) who were also in that brigade were originally equipped as and operated as dragoons--by the time of the Russo-Turkish War of 1853, they were considered heavy cavalry.
Note that this officer of the Fifth Dragoon Guards (who participated in Scarlett's "charge of the Heavy Brigade") is equipped with the straight military long sword, rather than a saber.