As to a cavalry division, in 1807 at Eylau, Grouchy acted just as you say.
When he arrived with his division of dragoons just as the Russians repulsed Augereau's infantry, Grouchy sent his 1st Brigade forward (during which charge his horse was wounded - he does not say he charged with them) and formed his 2nd Brigade to support the 1st Brigade.
His report in in volume 2 of his memoirs at starting on page 298. When visiting the USA, I could see the full volume on Google, yet only have a snippet view in Europe as it was published in 1873. - R