Dragoons rode to the site of the battle, dismounted, and only then engaged the enemy on foot. Their mounts were transportation, not "fighting vehicles."
Exactly why they are a pretty good analogy to Marines or Air Cavalry today: such forces possess no real fighting vehicles, just transportation vehicles with varying degrees of fighting shortcomings, pretty much how dragoons left the charging to those with proper (rare and expensive) warhorses and concentrated on other things such as raiding or holding. *
Starfleet Dragoons wouldn't need to go to battle on foot (even though with transporters, that's probably a valid type of tactical and strategic mobility), but their vehicles and craft wouldn't be real frontline "warhorses" or "tanks", either. Which would allow a TV episode to show a deployed SF Dragoon force by placing the usual eight extras in costumes in a dimly lit "outdoors" set, a cheapo location, or a rudimentary virtual environment, with the vehicles and craft done in miniature or background CGI or left out - while letting the depiction of actual vehicle rides or occasional vehicular fighting, with the related full-scale props and whatnot, to bigger-budget movies...
*) Of course, the glory of charging was too tempting, and eventually all dragoon forces began engaging in it. And as every Young Indiana Jones viewer remembers, 20th century Mounted Infantry sometimes succumbed to the temptation as well... I trust that SF Dragoons would be doing a lot of stuff beyond the real capabilities of their vehicles, too!