Yes, indeed. The Cavalryman was nothing without his horse but it is a frequent and expected thing that his horse would die. One of two things would happen.

The first expected item would be that he would make his way back to the encampment and hope that there was a horse avaiable for replacement. He would be vunerable to being randomly picked off by enemy light cavalry if they made it past the main army but that would be rare.

The other option would be that they would have to walk with the army. This often happened during the retreat in 1812 and in 1813 when there were not enough horses for all of the cavalry. Napoleon tried to mend this by creating foot versions of cavalry, such as a Dragoon a pied that existed for barely a few months.

I would recommend Swords Around A Throne By John Elting for a compositional history of the French Napoleonic army.

Typically, cavalry units had remount services who could provide replacement horses.