The response to crossbows appears to have been to make better armour. When powerful crossbows were common, the best armour was crossbow-proofed (i.e., successfully tested against crossbows).

The response to firearms was first, do nothing, since crossbow-proof armour stopped (i.e., had a good chance of stopping) bullets from early/mid 16th century hand-held firearms, with muzzle energies typically below 1000J. See https://www.reddit.com/r/AskHistorians/comments/5udybq/how_effective_was_plate_armor_against_musketballs/ for some past discussion by /u/WARitter of the penetration of armour by early firearms. For more, see Alan Williams, The Knight and the Blast Furnace, Brill, 2003.

Second, as guns improved, make the armour thicker (which also makes it heavier). Bullet-proof (i.e., successfully tested against bullets - typically pistol bullets) armour varied in thickness. Examples from about 3mm thick to about 9mm thick are found. See A. Williams, D. Edge, T. Atkins, "Bullet dents - "Proof marks" or battle damage", Gladius 26, 175-209 (2006) for examples.

Third, as the increase in thickness made the armour too heavy, get rid of the armour. The infantry was the first to discard armour. Cavalry retained bullet-proof armour for longer, and even when no longer wearing full armour, might still wear a bullet-proof breastplate. Siege engineers and specialist siege assault troops also retained heavy bullet-proof armour, in some case armour that would stop musket balls. These heavy siege armours were too heavy for regular wear on the battlefield, but in very high risk environments, could be useful.

Where does the Kelly Gang's armour fit into this? The armour was 4.5mm to 6mm thick (sources vary - /u/CChippy gives 3/16" in https://www.reddit.com/r/AskHistorians/comments/62iag9/how_thick_was_ned_kellys_armour/ and other sources often give 1/4"). Note that this is in the thickness range of bullet-proof armours of the 16th and 17th centuries.

Black powder pistols (see preceding link to past discussion of Kelly armour) of the type used against the armour typically had muzzle energies of about 300J. The rifles in use might have muzzle energies of about 1,500J, which would challenge the armour. Supposedly, the first version of the armour was found to not be strong enough, and thicker armour was made. From Williams, The Knight and the Blast Furnace, about 4.5mm of low-carbon steel armour (moderate quality) is required to stop about 1,500J.

The Kelly Gang were not facing muskets designed to penetrate armour on the battlefield (which could deliver in excess of 3,000J). This helped a lot. Even with this moderation of the threat they faced, their armour (at about the minimum thickness needed to stop bullets from the rifles they faced) was very heavy. Ned Kelly's armour was 44kg (97lb) (the other armours worn by the gang were lighter but less protective - Ned's armour had shoulder protectors and a rear apron, which the others lacked). This would have been rejected in the 17th century as too heavy for other than specialist siege/assault tasks.

In WW1, armour intended to stop bullets was used. One example, the German brow plate, or Stirnpanzer was used to reinforce the German steel helmet to protect against bullets (the helmet, by itself, was not bulletproof). The Stirnpanzer was 5mm thick, about the same as the Kelly armours.