This one has always nagged me. Considering that gunpowder did not comprise the majority of armies until the mid-17th century, why did large-shield tactics like the testudo formation (Roman era) or shield wall (Saxon/Viking era) come into less usage in the Medieval and Late Medieval periods (from about 1066 - 1600)? Correct me if I'm wrong, but when I think of shields from the latter periods, I think of small kite shields and bucklers used by pikemen.
Large-shield formations seem to have been very effective in protecting soldiers from the damage of missile projectiles, and would have been very useful, for example, for the French in the battle of Agincourt against the English longbows. Is it even a fair thing to state that shield size and usage seemed to decrease dramatically after 1066?
Now, I'm not a professional historian, and I don't have a lot of time to make this post great but I hope it will suffice.
Moving forward from the late classical period to the late medieval period or early modern period, in Europe, especially the more warlike regions such as Italy and western Germany, you see a general trend away from sword-armed formations of infantry to spear or pike armed infantry, and this is because it's easier to learn how to jab with a pike than to learn all these forms with swords, and also easier to manage in a formation, and also because it allows a deeper formation to make use of more than just the first rank of its formation. In other words, for the time period, it was a trend from a less efficient to a more efficient system.
This ties into the shields because large square shields are more obtrusive than the smaller pike buckles that your Italian pike squares would be using, and in many cases would be either neutralized due to its position if you try to hold the pike in both hands, or it would be simply impossible to manage that. Keep in mind that these new pikes and bill hooks and glaives and so forth are much more top heavy than the lances and pikes as used by the classical pikemen of antiquity, your hoplites and so forth, and so it's pretty necessary to hold it that way, with both arms.
I'm not so sure that shields universally became smaller. The Elizabethan target was not necessarily a very small thing, and those who still used them often still relied on a page or servant to carry them outside of combat. Even large rectangular shields still saw use in the middle ages, as seen in this 14th century Italian illustration.
What does happen in the later middle ages is that we start to see a growing preference for large, two-handed weapons, although how much of this is due to improvements in armor or a broader cultural shift I'll have to leave to someone else. Small bucklers could serve as a convenient sidearm hanging from a belt, as was popular among English longbowmen. They also may have simply been better in some situations. George Silver claimed that in 1 on 1 combat a sword and buckler had the advantage over a sword and target, but that in massed combat the target was better for defending against blows.
At Agincourt the French men at arms were likely pretty well protected against the English arrows. The main effect of the English Archers was to coax the French into charging prematurely before they were fully organized, and then wear them down as they slogged through the mud so that when they finally reached the English lines they were too exhausted to put up much of a fight.