Damage bonus is better than ASI per unit.
Say you had an option of ASI Med or Damage Increase: Med. The Damage Increase increases the DPS by more than the ASI, so go with the damage bonus.
Of course, ASI High against ADI Low (Swifty vs. Twisted Targe) is better, but if they were equal, then damage would be better.

This, however, depends on which weapons you use. For example, a Troika is better with ASI than damage bonus, even though it doesn't maximize its DPS potential. Just saying.

Interesting, though - there is a global damage bonus on the Chaos set.

The Swiftstrike is only really viable in T2, but if you can dodge LIKE A BAWS in T3, it's viable for a gunner. See also: Trinkets for ASI (or damage).

If you were to use trinkets for the opposite, at 5* for each, you could have ASI and Damage bonus VH - in which case, get whichever set defends against whatever bothers you most (for me? Devilites, so shadow armour) and the opposite trinket.

Juances has a point on the UVs, though. Hope to roll ASI, and get damage armour/trinkets, could help to get both. You can't roll damage vs everything UVs, like Juances said.

Incrased? Were are we? Deep south USA?

The best variants to use for gunner or for swords, from what i can find, is a combination of damage and speed, and trying to get them as close together as possible. Each level of ADI adds 5%, and each leve of ASI adds 4% speed, therefore 4% damage. Because these are separate bonuses to damage however, they are multiplied together. An easy example with swords is skolver cap and skolver coat, with swiftstrike buckler. Two medium increases of damage, or 20%, and one high increase of attack speed, or 12%. These multiply to 34.4% (1.2 * 1.12 = 1.334) total increase in dps. This is better than pure ADI, which caps at 30% and better than ASI, which caps at 24% speed increase. Not sure what items to use to reach that 4ADI 3ASI value for guns...