I'll take a stab at this: Archers would be annihilated vs. guns at several hundred yards where the archers could not reach effectively. Yes, the individual gun wouldn't hit squat at that range at that time, but a volley would...same as archers, but out of range of the archers. If you look at continental tactics you realize that tight ranks of musket armed men became quite deadly. Who wants to march point blank into a wall of musket fire at 3 to 4 rounds per minute per gun? It doesn't have to be aimed, it is fired as a volley to sweep an expanse on a relatively flat trajectory (something archers can't do.) Then there was artillery...who wants to engage artillery with archers? You will never get in range.

The American Civil War marked the apex of this arms race of formation fighting. Rifled cannon and repeaters led to trench style warfare late in the war (with mounted infantry to flank dug in positions--while saber wielding cavalry became obsolete since they were merely big, juicy, easy targets and could not dislodge entrenched infantry.) Rifled cannon are extremely accurate, have great range and can destroy hardened walls with ease, but are ineffective at destroying berms.

Since I mentioned cav, Nathan Bedford Forrest armed many of his men with revolvers to fight as dismounted infantry in the thickets. At short range in a thicket (where he fought) these were much more effective than rifles/carbines. Imagine facing some yahoo with dual revolvers at 20 paces in the woods while you have a single shot musket, breach loader, or carbine. If you've ever lived, hunted, or played paintball in forest thicket you can appreciate the typical engagement range you would face. Forrest also discarded the continental habit of keeping a large reserve (~1/3rd of your force). He threw everything he had at what he considered the decisive point. He understood that concentration of forces at the point of combat often decided the battle by breaking a key position. That (plus his leadership at the front) is why he was so effective with much smaller forces. Who cares about the reserve when your main body has been routed? All the reserve can do is halt your precipitous retreat and prevent a rout, not win the battle. On the flip side, should you fail to break a key position, your dismounted troops can still withdraw to their held horses, and retreat without being destroyed--if heavily bloodied. Defenders don't pursue immediately, since they must be wary of a trick or trap.