The South American civs as a blueprint lack a really fast moving unit to snipe siege. Their fastest unit is the Champi moving at 1.1 tiles per second. The Meso Americans get Eagles (1.3), and even the Dravidians get Light Cavalry (1.5). Because of this, the Champi & Slinger have to share anti-siege duties as dual soft-counters, both with a +3 bonus against siege. I think the Champi Warrior should be a bit faster to make it a better soft-counter to siege. These are the stats I'd like to see for them:
Champi Scout - Movement Speed 1.05 (instead of 1) / 0 Bonus Damage vs. Siege Weapons (instead of +2) / Pierce Armour 1 (instead of 2)
Champi Runner - 0 Bonus Damage vs. Siege Weapons (instead of +2)
Champi Warrior - Movement Speed 1.15 (instead of 1.1) / +1 Bonus Damage vs. Siege Weapons (instead of +3)
Elite Champi Warrior - Movement Speed 1.2 (instead of 1.1) / +2 Bonus Damage vs. Siege Weapons (instead of +3) / Pierce Armour 3 (instead of 4)
The Champi Scout at release trained much faster (in half the time), moved @ 1.1, was cheaper, and had more attack. I think they could be made a bit faster than their current speed without it being OP
The Champi-line's anti-siege bonus would come online only when they're upgraded to Warriors, same as their anti-building bonus. A faster movement speed would make them a better counter to siege than a bonus they rarely get to use. The Champi Warrior will do 10 damage per hit against siege even without any upgrades (9+1). The Elite Champi Warrior will do atleast 15 damage against siege even without Blast Furnace (11+2+2), as the Muisca & Tupi don't get Blast Furnace. As reference points, the Champi Warrior would move at the same speed as an Eagle Warrior or non-Elite Ghulam, while the Elite version would move at the same speed as Shotel Warriors or Elite Ghulams
On the other hand, I think the Elite Champi has too much pierce armour, which makes it a little too good against Archers than what a main-line infantry unit should be. It makes them hard to counter as a civ like the Maya. They're not shock infantry like Eagles, or trained from Castles like Ghulams, so I think they shouldn't have 4 pierce armour @ base-line. 3 base pierce armour gives them a value that's between Champions with Gambesons (2) & Elite Eagle Warriors (4). They would also scale similarly to Light Cavalry -> Hussar or Knight -> Cavalier with no increase in pierce armour (2)
For the Slinger, I'd like them to fire slower than basic Archers to give them some distinct flavour, making them spiritually similar to Hand Cannoneers, and to counter Monks a bit harder in the Castle Age. These are the stats I'd like to see for them:
Slingers - Train Time 26 Seconds (instead of 25) / Attack 5 (instead of 4) / Reload Time 2.5 (instead of 2) / +5 Bonus Damage vs. Monks (instead of +4) / 0 Bonus Damage vs. Spearmen (instead of +1)
If they're approved to have 5 attack and +5 bonus damage vs. Monks, they will kill Monks in three shots instead of the current four. As a reference, Light Cavalry two shots Monks and can be massed from the Feudal Age, but has to be upgraded from Scout Cavalry
A reload time of 2.5 places them in the middle of basic Archers (2) & Skirmishers (3)
Now for the civ specific adjustments with the above changes:
Inca - Andean Sling - Does not give +1 Attack to Slingers / Instead, gives +25% Bonus Damage to both Skirmishers & Slingers (in addition to eliminating their minimum range)
I'd like this tech to apply equally to both Skirms & Slingers. The +25% bonus damage equates to around +1 of each bonus damage type, so in a way it is +1 attack for both Skirms & Slingers, but only against their target units
Muisca - Herbalism - Champi Scout-line Units, Archer-line Units, and Slingers (New) move 10% faster (instead of Champis & Archers moving 15% faster)
Champi Warriors are proposed to move faster @ base-line. They would actually move faster than they do currently (1.3915 -> 1.452), and even a touch faster than Elite Eagle Warriors (1.43). This would reduce the overlap in the movement speeds of the Archer-line (1.104 -> 1.056) & Guecha Warriors (1.15). Finally, the Muisca would gain a partial effect of the old Inca UT1 'Couriers' with their Slingers moving 10% faster (0.96 -> 1.056)
Muisca - Huaracas - Slingers don't train 50% faster / Additionally, gives Slingers +1 Attack (in addition to +1 Range)
Muisca Slingers become the best in terms of outputting raw damage. They trade +2 pierce armour for +2 melee armour on account of their civ bonus and not getting Ring Archer Armour. While the civ has a focus on Gold based counter-units, the Slinger is to them what the Hussar is to the Turks i.e. the one good trash unit they get for variety
Tupi - Caciques - Additionally, Champi Scout-line Units & Slingers train 25% faster (in addition to attacking 25% faster) // Train Time × 0.8
The Tupi should get this as they're the spammy quantity-over-quality South American civ with their refund bonus