Dragoons (and 'mounted infantry' in general) function largely depended on who was using them. Even their role within the cavalry tended to vary. The Russians employed them as medium cavalry, using them for situations where Hussars were too light, but which didn't warrant attention from the cuirassiers.
I think the Austrians continued to use Dragoons in the mounted infantry role up until WWI, but I could be wrong. That would be uncharacteristically intelligent for the Austrian military leadership.
Cossacks were well known to act as mounted infantry when the situation warranted. Some hosts, like the Don, weren't willing to give up their horses, but the ones with less of a reputation were more than willing to run around on foot if it meant more loot at the end of a fight.
Very practical people, Cossacks.
For Grenadiers, Fisherking is exactly right. Grenadiers in the 18th century and onwards were the heavy infantry. They were the tallest men available and every measure was taken to up their intimidation factor, although I gather that sorting mustaches by unit soon fell out of favor on account of making everybody look silly.
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