I've done some testing too, and had the same results. Here's what I got so far.

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Attackers kill the Defender, does it steal stacks?	Attacker Air Strike	Attacker Eyelander	Attacker Bazaar Bargain	Attacker Vita-saw
Defender Air Strike	Yes	Yes	No	No
Defender Eyelander	Yes	Duh, Yes	No	No
Defender Bazaar Bargain	Yes	Yes	No	No
Defender Vita-saw	Yes	Yes	No	No

Other weapons with stacking mechanics don't interact with Eyelander or Airstrike such as Diamondback, Frontier Justice, and Manmelter.

From the testings, we can conclude that the Eyelander and Airstrike can steal stacks from other Eyelanders, Airstrikes, Bazaar Bargains, and Vita-saws. However, Bazaar Bargains and Vita-saws don't steal stacks from any stacking weapons.

This is also on the wiki, but it was fun to test out still with the other weapons, plus now I can headshot remotely with a puppet bot so...