During the Medieval period, knights and other men-at-arms routinely dismounted during siege operations. Although knights are popularly thought of as mounted warriors, they also trained vigorously for dismounted combat (although the popularity of dismounted combat in actual battles would ebb and flow over the period). Indeed there's a wealth of sources covering dismounted knightly combat throughout the Medieval period. For example, Hans Talhoffer's 1476 Fechtbuch covers a variety dismounted fighting techniques. And this is just one of many similar Fechtbücher, the earliest of which date back to the early 14th century. Dismounted fighting was also a fixture at medieval tourneys, as shown by illustrations like this drawing from the late 15th century Beauchamp Pageant and this Belgian illustration from 1530.

There are hundreds of examples of knights, squires, and other men-at-arms fighting dismounted during sieges.

Here are just a handful of examples:

During the Second Crusade's Siege of Damascus (July 1148), French and German forces approached the western part of the city. As the Crusaders moved through the orchards outside the city to begin siege operations, they came under attack. William of Tyre's account describes what happened next:

The cavalry forces of the townsmen and of those who had come to their assistance realized that our army was coming through the orchards in order to besiege the city and they accordingly approached the stream which flowed by the town. This they did with their bows and ballistas so that they could fight off the Latin army, which was fatigued by its journey and also so that they could prevent the thirsty men from reaching the river and the water which was so necessary for them. Our men hurried to the river, which they had heard was nearby, in order to relieve their thirst, which bad grown intense from the difficulties of their labors and the dense clouds of dust which were raised by the feet of horses and men. There they saw such a multitude of the enemy that they halted for a time. After a while they collected their men. They were given strength and hardiness by necessity. Once and then again they strove to get to the water, but in vain. While the king of Jerusalem and his men struggled vainly, the Emperor, who commanded the formations in the rear, demanded to know why the army was not moving forward. He was told that the enemy had seized the river and that they were blocking the progress of our men. When be learned of this, the Emperor was angered and, together with his lieutenants, he speedily made his way through the French King's ranks to the place where the fight for the river was going on. They dismounted from their horses and became infantrymen-as the Germans are accustomed to do in the crisis of battle. With shields in hand they fought the enemy hand-to-hand with swords. The enemy, who had earlier resisted valiantly, were unable to withstand the attack. They relinquished the river bank and fled at full speed to the city.

The other sieges of the Crusades likewise saw knights fighting dismounted. At Acre (1189-1191), dismounted knights taking part in the Third Crusade took part in several unsuccessfully attempts to storm the breaches in the city's walls (although the city was eventually forced to surrender. During the Second Siege of Acre (1291), dismounted knights would be on the defensive--the Templar of Tyre would write of fierce, but ultimately futile resistance by dismounted knights defending the walls and towers of the town from Saracen attack.

Mounted men-at-arms could also be used during sieges. It on the offensive, they might be held as a mounted reserve to counter any enemy relief forces or enemy sallies from inside the castle/town. On the defensive, they used as reserves to counter any enemy or to harass enemy soldiers outside the walls. Stretching the timeline a bit, the Siege of Malta in 1565 has some good examples of this. Maltese cavalry fought brisk, highly successful skirmishes against Turkish infantry foraging in the countryside.

As armored men faded from the battlefield gunpowder became more prevalent, siege tactics became more and more about the use of earthworks and artillery. It became rarer for cavalry to dismount and fight in sieges. Instead, cavalry was used to screen the siege force (to ensure it wasn't surprised from the rear) and served as a mounted reserve. If enemy troops tried to sally from the fort to attack the fatigue parties digging trenches, the reserve force of cavalry and infantry would swiftly counterattack.