Carrhae would be the best example, I think, of cavalry tactics against infantry, and a good illustration of the combined use of light and heavy cavalry - the Parthian horse archers, keeping out of range of the Roman missiles, kept up a steady fire against the Roman infantry, causing them to bunch together for protection and so buckling their line. These gaps or weak spots in the Roman line were then attacked by small bodies of heavy cataphracts, acting more like wedges or small columns, I would think, than the charging wall of popular imagination. The cataphracts were driven off, but the steady attrition wore the Romans down and led to their defeat.
In 38BC, Roman general Ventidius again faced the Parthian cavalry - this time he positioned his men at the top of a slope, which either meant the horse archers were unable to shoot effectively (the source for this, Frontinus, is a bit vague), or made it appear that the Romans were already in disarray. Either way, an uphill charge by the Parthian cavalry was repelled by a downhill counter-charge by the Roman infantry, resulting in the total destruction of the Parthians.
An example of a battle apparently entirely fought by cavalry occurs in Zosimus' description of the Palmyran campaign of Aurelian:
Quote:...observing that the Palmyrene cavalry placed great confidence in their armour, which was very strong and secure, and that they were much better horsemen than his soldiers, he [Aurelian] planted his infantry by themselves on the other side the Orontes. He charged his cavalry not to engage immediately with the vigourous cavalry of the Palmyrenians, but to wait for their attack, and then, pretending to fly, to continue so doing until they had wearied both the men and their horses through excess of heat and the weight of their armour; so that they could pursue them no longer. This project succeeded, and as soon as the cavalry of the emperor saw their enemy tired, and that their horses were scarcely able to stand under them, or themselves to move, they drew up the reins of their horses, and, wheeling round, charged them, and trod them under foot as they fell from their horses. By which means the slaugther was promiscuous, some falling by the sword, and others by their own and the enemies' horses. After this defeat, the remains of the enemy fled into Antioch...
Zosimus - Historia Nova I
It's unclear quite what the Palmyrene cavalry thought they were doing here - they were perhaps trying to outflank the infantry, but the latter were on the far side of a river...
It's possible this is the same battle as the second clash at Emesa described by Zosimus - the dubious Historia Augusta gives only one major confrontation. In any case, in this second example the previous plan seems to go awry:
Quote:At the commencement of the engagement, the Roman cavalry receded, lest the Palmyrenes, who exceeded them in number, and were better horsemen, should by some stratagem surround the Roman army. But the Palmyrene cavalry pursued them so fiercely, though their ranks were broken, that the event was quite contrary to the expectation of the Roman cavalry. For they were pursued by an enemy much their superior in strength, and therefore most of them fell. The foot had to bear the brunt of the action. Observing that the Palmyrenes had broken their ranks when the horse commenced their pursuit, they wheeled about, and attacked them while they were scattered and out of order. Upon which many were killed, because the one side fought with the usual weapons, while those of Palestine brought clubs and staves against coats of mail made of iron and brass. The Palmyrenes therefore ran away with the utmost precipitation, and in their flight trod each other to pieces, as if the enemy did not make sufficient slaughter; the field was filled with dead men and horses...
Zosimus - Historia Nova I
In this case, the Palmyrenes were definitely trying an outflanking movement against the infantry. The retreat of the Roman cavalry might have been feigned, as before, with the intention of drawing off the Palmyrene horse and allowing the infantry to manoeuvre against them - the wheeling of the Roman infantry line presumably brought them into contact with the open flank of the Palmyrene cavalry advance: a rare case of infantry outflanking cavalry, perhaps?