They did indeed; mounted infantry were a major component of later medieval English armies. Both archers and men-at-arms during the Hundred Years War served as mounted infantry; my answer will concern the period of occupation between 1417-1450 (though the mounted archer came to prominence during the 1330s and 40s). These soldiers would have dismounted to fight, the archers obviously, but the men-at-arms too would have dismounted, for example, at Verneil, where the only contribution of actual cavalry was being handily driven off by a rearguard of English archers.
As a general rule, mounted archers would be used for field service and those without mounts would be used in garrisons. The division between field vs. garrison service was pretty solid in this period, with very few of the soldiers who served in France repeatedly moving between the two.
That said, there certainly were some mounted soldiers serving in garrisons, for example in the first 1427 Harfleur muster 18 of the 47 archers are recorded as mounted. In 1435 Rouen muster roll 8 of the 44 men-at-arms are listed as mounted. There was a good deal of utility for this; the English forces made efforts to suppress brigands during this period and mounted soldiers would be more effective in this role. Furthermore, mounted infantry present in garrisons could be broken off from the garrison to serve in the field (John, duke of Bedford and regent of France instituted a policy of having detachments in garrisons for this very purpose in 1434).
The distinction between foot and mounted archers seems largely to have been in their military function; both types of archer, serving in garrisons, were paid the same wages. Foot archers come to prominence later than their mounted archers, appearing with greater frequency once the Crown had more garrisons to maintain.
In contrast to the archers, foot men-at-arms serving in garrisons received much lower wages than mounted men-at-arms. It's entirely possible that the move from foot man-at-arms to mounted was seen as a promotion, given the higher wages.
Further reading:
The Soldier in Later Medieval England by Bell et al. is enormously valuable for this sort of thing.
There are. The idea of mounted infantry was known in ancient Egypt and Mesopotamia (both for horse riders and charioteers). In the Middle Ages, the practice was still in use, as at least small part of the Frankish cavalry under Charles Martel fought on foot at Poitiers [1].
In the late Middle Ages/early Rennaisance (that are more or less the same period, depending on the region) mounted infantry was not too popular yet it was by all means present. It is argued that the Irish and English hobelars were acting as either light cavalry of mounted infantry, depending on the circumstances [2][3]. Geonese crossbowmen were also known to use the horses for transport, yet they fought almost universally on foot.[4] Hussite mounted crossbowmen were also mounted infantry out of necessity, as majority or forces, especially Taborites, recruited from commoners who had access to draw and pack horses that performed poorly in combat. This is why Hussite cavalry was most commonly used for reconnaissance and transport.[5]