At the Civil War battle of Brandy Station, the attacking Union cavalry general Alfred Pleasanton brought along some 3,000 infantry to even the odds, because his cavalry was outnumbered 9,500 to 8,000 by that of Confederate commander J.E.B. Stuart. During the battle, both sides were forced to dismount (with one quarter of the cavalrymen holding the horses of three others), so the 3,000 infantry was actually worth 4,000 cavalry when this happened. (The Confederates won a tactical victory, but the Union troops gained valuable fighting experience.)
Brandy Station showed what infantry could do against cavalry in an era of "repeating" weapons such as Colt revolvers and breech-loading rifles. But did it make sense for a qualitatively and quantitatively inferior cavalry force to reinforce itself in this way using infantry to increase your numbers in attacking enemy cavalry before the introduction of fireams? That is to say, in the ancient period or Middle Ages, when your infantry was likely to be inferior to your cavalry in fighting enemy cavalry? I'm not talking about situations in say the 100 Years' war at Crecy or Poitiers when English infantry and cavalry fought on the defensive, while archers did the damage. A better example would be at Alesia: Should Vercingetorix have sent infantry along with his cavalry to try to disrupt the Roman circumvallation process (which was fended off by Caesar's German cavalry)?
Some other potential applications: Are there times when attacking cavalry can use infantry to defend key points such as passes? Can infantry be used profitably to "flush out" enemy cavalry from defensive ground such as trees and bushes, and into the open field? Can infantry be used as a rear guard against enemy cavalry while your cavalry regroups for a counterattack?
Or were the disadvantages of infantry against cavalry before the days of firearms so great as to preclude such uses?